CHARACTERISTIC CLASSES
AND SINGULAR VARIETIES
Jean-Paul BRASSELET
CNRS Marseille
Preamble
These notes are prepared for a course given during the School on Singularities in Geometry,
Topology, Foliations and Dynamics 2014, A celebration of the 60th birthday of Pepe Seade.
A first version of the course was given during the XVII Encontro Brasileira de Topologia,
August 2nd to 6th, 2010 in PUC, Rio de Janeiro, Brasil. Notes of this course are partially
taken from a book in preparation. Thanks to the reader for providing comments, critics
etc... in order to improve the final version.
The first three chapters are covered as well by the course given by Pepe Seade. They
are provided in order to fix notations and results that are useful for the following.
Contents
1 Euler-Poincaré characteristic
1.1 Combinatorial definition . . . . .
1.2 Manifolds - Poincaré isomorphism
1.3 Pseudomanifolds . . . . . . . . .
1.4 The genus of surfaces . . . . . . .
1.5 Betti numbers . . . . . . . . . . .
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6
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2 Poincaré-Hopf Theorem (smooth case)
2.1 The index of a vector field. . . . . . . . . . . . .
2.2 The index - Definition by obstruction theory . .
2.3 Relation with the Gauss map . . . . . . . . . .
2.4 Poincaré-Hopf Theorem . . . . . . . . . . . . .
2.4.1 The smooth case without boundary . . .
2.4.2 Consequences of Poincaré-Hopf Theorem
2.4.3 The smooth case with boundary . . . . .
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13
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3 Characteristic classes : the smooth case
23
3.1 Fibre bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.1 Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.2 Fibre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1
3.2
3.3
3.4
3.5
3.1.3 Examples of fibre bundles - real case . .
3.1.4 Examples of fibre bundles - complex case
General obstruction theory . . . . . . . . . . . .
3.2.1 The difference cochain . . . . . . . . . .
3.2.2 The obstruction class . . . . . . . . . . .
Case of the tangent bundle . . . . . . . . . . . .
3.3.1 Index of a r-frame . . . . . . . . . . . .
Applications: Stiefel-Whitney and Chern classes
3.4.1 Stiefel-Whitney classes . . . . . . . . . .
3.4.2 Chern classes . . . . . . . . . . . . . . .
Axiomatic definition . . . . . . . . . . . . . . .
4 Hirzebruch theory
4.1 The arithmetic genus . . . . . . . .
4.2 The Todd genus . . . . . . . . . . .
4.3 The signature . . . . . . . . . . . .
4.4 Hirzebruch Theory . . . . . . . . .
4.4.1 Hirzebruch Series . . . . . .
4.5 Characteristic Classes of Manifolds
4.6 The χy -characteristic . . . . . . . .
4.7 Hirzebruch Riemann-Roch Theorem
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5 Singular varieties
5.1 Stratifications . . . . . . . . . . . . . . . .
5.1.1 Whitney stratifications . . . . . . .
5.2 Poincaré homomorphism . . . . . . . . . .
5.2.1 Alexander isomorphism . . . . . . .
5.3 Poincaré-Hopf Theorem: The singular case
5.3.1 Radial extension process . . . . . .
5.3.2 Poincaré-Hopf Theorem for singular
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varieties.
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6 Schwartz and MacPherson classes
6.1 Radial frames . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Global radial extension . . . . . . . . . . . . . . . .
6.2 Schwartz classes . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Nash transformation . . . . . . . . . . . . . . . . . . . . .
6.4 Mather classes . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Euler local obstruction . . . . . . . . . . . . . . . . . . . .
6.5.1 Properties of Euler local obstruction . . . . . . . .
6.6 MacPherson classes . . . . . . . . . . . . . . . . . . . . . .
6.7 Schwartz and MacPherson classes . . . . . . . . . . . . . .
6.8 Schwartz-MacPherson classes for projective cones . . . . .
6.9 Schwartz-MacPherson classes of Thom spaces associated to
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embeddings
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7 Other classes and comparisons
7.1 Fulton classes . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Milnor classes . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1 Description in terms of constructible functions . . . . .
7.3 Motivic Chern classes: Hirzebruch theory for singular varieties
7.4 Verdier Riemann-Roch Formula . . . . . . . . . . . . . . . . .
3
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73
Introduction
The Euler-Poincaré characteristic is the first characteristic class that has been introduced. The Poincaré-Hopf theorem says that, if X is a compact manifold and v a continuous vector field with a finite number of isolated singularities ak with indices I(v, ak ),
then
X
χ(X) =
I(v, ak ) .
That means that the Euler-Poincaré characteristic is a measure of the obstruction to the
construction of a non-zero vector field tangent to X.
Later on, Severi and Todd (1935) defined characteristic cycles of projective varieties
using polar varieties.
In his famous paper, Chern (1946) defined characteristic classes for hermitian manifolds in several ways, in particular as the measure of the obstruction to the construction
of complex r-frames tangent to the manifold, generalising the Poincaré-Hopf theorem.
The Chern classes are represented by algebraic cycles which coincide with Todd cycles.
During several years, the attractiveness of the axiomatic properties of Chern classes
caused the viewpoints of obstruction theory and polar varieties to be somewhat forgotten.
It is interesting to see that these viewpoints came back on the scene with the question of
defining characteristic classes for singular varieties.
There are in fact various definitions of characteristic classes for singular varieties. In
the real case, there is a combinatorial definition, which simplifies the problem. In the
complex case, the situation is more complicated (and certainly more interesting !), due
to the fact that there is no combinatorial definition of Chern classes. The obstruction
theory point of view, in the smooth case, is based on the existence of the tangent bundle.
If one wants to use the obstruction theory point of view in the singular case, one has to
find a substitute to the tangent bundle. There are various candidates to substitute the
tangent bundle and each of them leads to a different definition of Chern class for singular
varieties. In particular, one has the following three substitutes.
a) If X is a singular complex analytic variety, equipped with a Whitney stratification
and embedded in a smooth complex analytic manifold M one can consider the union of
tangent bundles to the strata, that is a subspace E of the tangent bundle to M . The
space E is not a bundle but it generalises the notion of tangent bundle in the following
sense: A section of E over X is a section v of T M |X such that in each point x ∈ X, then
v(x) belongs to the tangent space of the stratum containing x. Such a section is called
a stratified vector field over X. To consider E as a substitute to the tangent bundle of
X and to use obstruction theory is the M.H. Schwartz point of view (1965), for defining
Chern classes of analytic complex varieties.
b) A second possibility is to consider, for x singular point of X, the space of all
possible limits of tangent vector spaces Txn (Xreg ) where xn is a sequence of points in the
regular part Xreg of X converging to x ∈ X. That point of view leads to the notion
4
of Mather classes, which are an ingredient in the MacPherson definition for classes of
algebraic complex varieties (1974). Another main ingredient for the definition of these
classes is the notion of Euler local obstruction that we study in Section 6.5.
c) A third possibility for a substitute to the tangent bundle is the following: Let us
suppose that there exists a normal bundle N to X in M , that is the case of local complete
intersections for example. Then, one can consider the virtual bundle T M |X \ N as a
substitute to the tangent bundle of X. That point of view is the one of Fulton (1980).
There are relations between the classes obtained by the previous constructions. First
of all, the Schwartz and MacPherson classes coincide, via Alexander duality (1979, [B-S]).
The relation between Mather classes on one side and Schwartz-MacPherson classes
on the other side follows form the MacPherson’s definition itself: His construction uses
Mather classes, taking into account the local complexity of the singular locus along Whitney strata. This is the role of the local Euler obstruction.
A natural question arised to compare the Schwartz-MacPherson and the Fulton classes.
A result of Suwa [Su1] shows that in the case of isolated singularities, the difference of
these classes is given (up to sign) by the sum of the Milnor numbers in the singular points.
It was natural to call Milnor classes the difference in the general case. This difference
has been described by several authors using different methods (P. Aluffi, J.P. Brasselet-D.
Lehmann-J. Seade-T. Suwa, A. Parusiński-P. Pragacz and S. Yokura).
In the case of manifolds, the Todd genus and the L-genus are degree 0 elements of
the Todd class and the Thom-Hirzebruch L-class. Both of them are related to the Chern
class via the Chern roots and F. Hirzebruch gave a way to unify these three theories
of characteristic classes by using the so-called multiplicative series. In the same way
that the MacPherson construction generalises the Chern class, the Todd class and the
Thom-Hirzebruch L-class have been generalised as natural transformations respectively
by Baum-Fulton-MacPherson and by Cappell-Shaneson. The problem is that the three
transformations are defined on different groups on the singular complex algebraic variety
X: namely group of constructible functions, Grothendieck group of coherent sheaves,
group of constructible self-dual sheaves. One way to unify the three theories, in the
singular case, is to use the motivic theory and the Grothendieck relative group of algebraic
varieties over X. That has been performed by J.-P. Brasselet, J. Schürmann and S. Yokura
and is explained in the section 7.3.
The author thanks the Scientific and Organising Committees of the School and Workshop on Singularities in geometry, topology, foliations and dynamics, Cuernavaca 2014,
for providing oportunity to give a course during the School.
Aubagne, August 18th, 2014
5
1
Euler-Poincaré characteristic
In this section, the varieties we consider are possibly singular varieties.
1.1
Combinatorial definition
History of characteristic classes begins with the discovery of the so-called Euler formula, by
Leonhard Euler around 1750 : Let P be a 2-dimensional polyhedron in R3 , homeomorphic
to the sphere S2 , one has
k0 − k1 + k2 = 2
where k0 is the number of vertices in P , k1 is the number of segments and k2 the number
of faces. That is the case for the tetraedron: 4 − 6 + 4, for the cube (with diagonals on
the faces): 8 − 18 + 12.
According to different authors, that formula was first proven by Euler himself, by
Legendre in 1794 or by Cauchy. In fact, it seems that this formula was already known by
R. Descartes (around 1620) and even by Archimedes.
Simon Antoine-Jean Lhuilier, a Swiss mathematician, gave (in 1812) a slight generalization of Euler’s formula taking into account orientable 2-dimensional polyhedra with
holes. The number g of holes is called genus (see Definition 1.16). Lhuilier’s formula is
k0 − k1 + k2 = 2 − 2g,
where g is the genus. Thus one obtains 0 for a torus-like polyhedron.
For a general 2-dimensional polyhedron P in R3 , the alternative sum
χ(P ) = k0 − k1 + k2
is called Euler characteristic of P .
H. Poincaré [Po2] generalized the result in 1893 for finite polyhedra P of higher dimensions and proved the so-called Poincaré-Hopf Theorem, which is the bridge to differential
geometry. One defines
Definition 1.1 Let us denote by ki the number of i-dimensional simplices of a finite
n-dimensional polyhedron P in Rm , the Euler-Poincaré characteristic of the polyhedron
P is defined by
n
X
χ(P ) =
(−1)i ki .
i=0
Let us remind some elementary definitions:
6
Definition 1.2 A (finite) simplicial complex K is a collection of simplexes in some euclidean space such that
• if s ∈ K then every face of s belongs to K,
• if s, t ∈ K, then s ∩ t is either empty or is a common face of s and t.
Definition 1.3 Let us denote by K a (finite) simplicial complex in Rm . The union of
simplexes in K is a compact subspace of Rm denoted by |K| = P and called geometric
realisation of K, or polyhedron associated to K.
Definition 1.4 A topological space X is triangulable (or a polyhedron) if there exists a
simplicial complex K and a homeomorphism h : |K| → X. Such a pair (K, h), or simply
the simplicial complex K, is called a triangulation of X.
The Poincaré’s result is the following:
Theorem 1.5 (Poincaré, [Po2]) Let us consider two triangulations (K1 , h1 ) and (K2 , h2 )
of a (compact) topological space X, then one has χ(K1 ) = χ(K2 ).
Theorem 1.5 implies that Euler-Poincaré characteristic is a topological invariant of the
space X. The result makes sense for the following definition.
Definition 1.6 The Euler-Poincaré characteristic of the triangulable space X, denoted
by χ(X) is defined as χ(K) for a triangulation (K, h) of X.
Examples 1.7 The Euler-Poincaré characteristic of the sphere is χ(Sn ) = 1 + (−1)n , of
the 2-dimensional real torus T is χ(T ) = 0, of the pinched torus is χ(T ) = 1.
The Euler-Poincaré characteristic of the complex projective space is χ(CPn ) = n + 1.
1.2
Manifolds - Poincaré isomorphism
Let us recall the definition of topological manifold:
Definition 1.8 (Topological manifold) A Hausdorff space is called a (topological) mmanifold if each point x in M admits a neighbourhood Ux homeomorphic to a ball Bm ⊂
Rm through a homeomorphism φ : Ux → Bm such that φ(x) = 0 and the boundary of Ux ,
called the link of x, is homeomorphic to the sphere Sm−1 .
Let us denote by M a m-manifold and by (K) a triangulation of M . A dual cell
decomposition of M is obtained in the following way:
Let us consider a barycentric subdivision (K 0 ) of (K). The barycenter of a simplex
σ ∈ K will be denoted by σ̂. Every simplex in K 0 can be written as
(σ̂i1 , σ̂i2 , . . . , σ̂ip )
where σi1 < σi2 < · · · < σip . Here the symbol σ < σ 0 means that σ is a face of σ 0 .
The dual cell of a simplex σ, denoted by d(σ), is the set of all (closed) simplexes τ in
(K 0 ) such that τ ∩ σ = {σ̂}. That is the set of simplexes on the form (σ̂, σ̂i1 , . . . , σ̂ik ) with
σ < σi1 < · · · < σik .
The dual cells satisfy the nice properties:
7
A0
L0
B0
C0
C0
K0
A
D0
A
B
C
C
C
G0
A0
C0
L0
K0
C0
C0
C0
A
B
A
H0
B
H0
(a) Dual of the triangle ABC
is the barycenter {C 0 }
G0
B
F0
E0
B
E0
D0
A
D0
B
B̃
E
F0
G0
C0
Ã
D0
E0
K0
A
B0
B0
L0
F0
E
C
A0
A
E0
H0
(b) Dual of AC is the 1E
dimensional cell {A0 B 0 C 0 }
(c) Dual of the vertex {A0 } is
the 2-dimensional cell Ã
C0
Ã
Figure 1: Dual cells
Ã
A
D0
C0
E0
B
B̃
0
Lemma 1.9 A 1. The Ddual
cell is a cell, homeomorphic to a ball and its boundary is
B̃
homeomorphic to0 the corresponding
sphere.
B
E
2. If σ is a k-simplex, then d(σ) is a (m − k)-cell.
3. The set of dual cells provide a cell decomposition of M , called dual cell decomposition
associated to the barycentric subdivision (K 0 ) of (K).
The unique intersection point σ̂ = d(σ) ∩ σ is the barycenter of σ that we will denote
ˆ
also sometimes by dˆ = d(σ).
Let us assume M = |K| oriented, that is all m-simplices are given a compatible
orientation. One gives to every cell d(σ) the orientation such that orientation of σ followed
by orientation of d(σ) is orientation of M .
Let us fix some notations:
• We denote by d∗ (σ) the elementary (D)-cochain whose value is 1 at the cell d(σ)
and 0 at other cells of (D).
(K)
• We denote by Ci the groups of simplicial K-chains with integer coefficients and
i
the groups of simplicial D-cochains with integer coefficients.
by C(D)
Let us consider a compact oriented m-dimensional manifold, then one has, for every
k, a chain isomorphism:
(K)
m−k
C(D)
(M ; Z) −→ Ck (M ; Z),
(1.10)
that one defines on the elementary elements as
d∗ (σ)
7→
σ
and extends linearly.
The following Theorem is one of the possible forms of the Poincaré duality:
Theorem 1.11 (Poincaré isomorphism) Let M be a compact oriented m-dimensional
manifold, the morphism (1.10) induces, for every k, an isomorphism
H m−k (M ; Z) −→ Hk (M ; Z) ,
which is the cap-product with the fundamental class [M ] ∈ Hm (M ; Z).
8
B
1.3
Pseudomanifolds
The spaces we will consider are pseudomanifolds. This corresponds to the spaces called ncircuit by Poincaré and Lefschetz. In fact, the notion of pseudomanifold differs according
to the authors.
Definition 1.12 (Pseudomanifold - Combinatorial definition) One says that the polyhedron |K| is an n-pseudomanifold if the simplicial complex K satisfies the following
properties:
(i) dim K = n, i.e. the maximal dimension of simplexes in K is n.
(ii) Each simplex is face of a n-simplex.
(iii) Each n − 1-simplex is face of exactly two n-simplexes.
The notion of “simplicial simple n-circuit” (Lefschetz [Le], Poincaré) corresponds to
the one of pseudomanifold with the following additional connexity property
(iv) The set of the n and n − 1-simplexes is connected.
The property means that |K| \ |K (n−2) | is connected. Equivalently, given two n simplexes
σ and τ in K, there exists a sequence of n-simplexes σ = σ1 , σ2 , . . . , σr = τ such that
σi ∩ σi+1 is an (n − 1)-simplex.
If properties (i) to (iv) are verified, we will say that |K| is a simple n-pseudomanifold.
The topological definition of pseudomanifolds, which is equivalent to the combinatorial
one in the case of triangulable topological space, goes as follows:
Definition 1.13 (Pseudomanifold - Topological definition) One says that the (paracompact, Hausdorff) topological space X is an n-pseudomanifold if there is a subset Σ ⊂ X
such that:
(i0 ) dim X = n.
(ii0 ) X \ Σ is a n-topological manifold dense in X.
(iii0 ) dim Σ ≤ n − 2.
The property (iv) is equivalent to the following connexity property
(iv0 ) The set X \ Σ is connected.
If properties (i0 ) to (iv0 ) are verified, we will say that X is a simple n-pseudomanifold.
In the triangulated case, one can take Σ = |K (n−2) |.
Example 1.14 The pinched torus, the suspension of the torus, a Thom space, a complex
algebraic variety are examples of simple pseudomanifolds.
Let K be a triangulation of a connected homology n-manifold (over Z), then |K| is
an n-pseudomanifold.
9
b
a
A
(a) Pinched torus. The link
of the singular point A is
union of two circles.
(b) Suspension of the torus.
The link of the vertex A is a
torus
(c) Suspension of the torus
(similar arrows should be
identified at each level)
Figure 2: Exemples of pseudomanifolds
Not all n-pseudomanifolds are homology n-manifolds. The pinched torus and the
suspension of the torus are pseudomanifolds, they are not (homological) n-manifolds.
Proposition 1.15 An oriented simple n-pseudomanifold X admits a fundamental class
[X] ∈ Hn (X).
Proof:
Let us consider a triangulation (K) of X. It is easy to verify that the sum
of oriented n-simplices is a cycle, called a fundamental cycle. Its homology class is the
fundamental class of X.
We show in section 5.2 that for an oriented pseudomaniflod, the cap product with the
fundamental class [X] defines a homomorphism
H n−k (X; Z) −→ Hk (X; Z) .
1.4
The genus of surfaces
Definition 1.16 The genus g of a connected surface is the integer representing the maximum possible number of cuttings along closed simple curves without obtaining a disconnected manifold.
Proposition 1.17 An orientable surface of genus g can be obtained from S2 by successively attaching handles g times.
Let us describe the “attaching process” used in Proposition 1.17: Let us consider a
connected surface M and an embedding f : S0 × D2 → M \ ∂M . Image of f is a pair of
disjoint disks in M . Cut out interior of these disks and glue in the cylinder D1 × S1 by
f |S0 ×S1 . One says that the resulting surface M 0 is obtained from M attaching an handle
by f :
M 0 = M \ Intf (S0 × D2 ) ∪f D1 × S1 .
Among orientable surfaces, genus of the sphere is 0, genus of the torus is 1. The
Lhuilier formula is χ(X) = 2 − 2g (Proposition 1.26).
For non-orientable surfaces, one has the following results :
10
Lemma 1.18 (see [Hirs]) Every non-orientable surface contains a Möbius strip.
Theorem 1.19 (see [Hirs]) For a compact connected non-orientable surface without boundary the genus is the unique integer g such that M contains g but not g + 1 disjoint Möbius
strips.
Proposition 1.20 A non-orientable surface of genus g ≤ 1 is diffeomorphic to the connected sum of g disjoint copies of RP2 , real projective plane.
The real projective plane RP2 is a non-orientable surface of genus 1. One has χ(RP2 ) =
1. The Klein bottle B is a non-orientable surface of genus 2, one has χ(B) = 0. The
sphere S2 with g real projective planes attached is a non-orientable surface of genus g.
We will see (Proposition 1.26) that if X is a non-orientable surface of genus g, then one
has χ(X) = 2 − g.
1.5
Betti numbers
In 1871, Betti [Be] defined numbers relative to 3-dimensional compact manifolds without
boundary and announced a duality property.
A first statement of Poincaré duality was provided by Henri Poincaré in 1893 in terms
of Betti numbers: The i-th and (n − i)-th Betti numbers of a closed (i.e. compact and
without boundary) orientable n-manifold are equal. In his 1895 paper “Analysis Situs”
[Po3], Poincaré proved the theorem using a new tool: topological intersection theory.
In his danish dissertation thesis, 1898, Poul Heegaard [He] (french translation in [HeF])
gives a counter-example to the version of Poincaré duality. The Heegard paper forced
Poincaré to be more precise.
In fact, Poincaré had overlooked the possibility of the appearance of torsion in the
homology groups of a space. According to Poincaré, the Heegard and Poincaré definitions
of Betti numbers (in fact definitions of homologies) are not the same (see NDLR, page 161
in [HeF]). The Betti definition does not take into account possibility to consider cycles
with coefficients.
In order to underline clearly this fact and to provide an indisputable proof, Poincaré
wrote the first complement to Analysis situs [Po4]. In the first two complements to
Analysis Situs [Po4, Po5], Poincaré gave a new proof in terms of dual triangulations.
Let us denote by |K| a n -dimensional finite polyhedron and by Ci (K) the finitely generated free abelian group whose generators are (oriented) i-simplexes of the triangulation
K. The boundary operator is classically defined as a complex map ∂i : Ci (K) → Ci−1 (K).
The subgroups of cycles and boundaries
Zi (K) = Ker[∂i : Ci (K) → Ci−1 (K)]
and
Bi (K) = Im[∂i+1 : Ci+1 (K) → Ci (K)]
are finitely generated, as subgroups of a finitely generated group. The homology groups
Hi (K, Z) = Zi (K)/Bi (K) are also finitely generated, as quotient group of a finitely generated one. One can write
Hi (K, Z) = Fi (K) ⊕ Ti (K)
where Fi (K) is the free subgroup and Ti (K) the torsion subgroup of Hi (K, Z).
11
Definition 1.21 The Betti numbers of |K| are defined as
βi (K) = rk (Hi (K, Z)) = rk (Fi (K)).
Equivalently, one can define the Betti number βi (K) as the dimension of the vector space
Hi (|K|; Q).
Noting that βi (K) = 0 if i > dim |K| = n, one has the Poincaré Theorem:
Theorem 1.22 (Poincaré Theorem) [Po3] Let |K| be a finite polyhedron in Rm , with
Betti numbers βi (K), one has
n
X
χ(K) =
(−1)i βi (K).
i=0
Proof:
From the long exact sequence
∂i+1
∂
i
· · · −→ Ci+1 (K) −→ Ci (K) −→
Ci−1 (K) −→ · · ·
one deduces the following equalities (where ni = rk (Ci (K)) and n = dim(|K|))
n
n
n
n
X
X
X
X
i
i
i
(−1) ni =
(−1) (dim Zi +dim Bi−1 ) =
(−1) (dim Zi −dim Bi ) =
(−1)i βi (K).
i=0
i=0
i=0
i=0
The first one comes from the short exact sequence
0 → Zi → Ci → Bi−1 → 0 ,
the second one because B−1 = ∅ and Bn = ∅ and the third one from the Definition 1.21.
The Theorem follows.
Alexander [Al] proved in 1915 that two triangulations (K, h) and (K 0 , h0 ) of the same
topological space X have same Betti numbers βi (K) = βi (K 0 ) for every i. One can define
βi (X) as being βi (K) for any triangulation (K, h) of X and one has
χ(X) =
n
X
(−1)i βi (X).
i=0
This result proves that each Betti number is a topological invariant. In that sense, it is
more precise than the Poincaré Theorem 1.5, which globally proves invariance of EulerPoincaré characteristic only.
Theorem 1.23 The Betti numbers of a compact orientable n-manifold M satisfy
βi (M ) = βn−i (M )
for i = 0, 1, . . . , n.
Corollary 1.24 If M is a compact orientable n-manifold with odd n, then χ(M ) = 0.
Examples 1.25 If n is odd, the Euler-Poincaré characteristic of the sphere Sn , the real
projective space RPn , a compact hypersurface in Rn+1 , are zero.
Proposition 1.26 If X is an orientable (connected) surface of genus g, then β0 (X) = 1,
β1 (X) = 2g and β2 (X) = 1. One has χ(X) = 2 − 2g.
In X is a non-orientable surface of (non-orientable) genus g, then β0 (X) = 1, β1 (X) =
g − 1 and β2 (X) = 0. One has χ(X) = 2 − g.
12
2
Poincaré-Hopf Theorem (smooth case)
The Poincaré-Hopf Theorem is important for many points of view: that is the first result
that links two invariants from topology and differential geometry.
The Poincaré-Hopf Theorem has been proved by Poincaré [Po1] in 1885, in the 2dimensional case, and by Hopf in 1927 [Ho] for higher dimensions. In between, partial
results had been proved by Brouwer and Hadamard. This result is the first apparition
of Euler-Poincaré characteristic in differential topology, out of combinatorial topology. It
seems strange that Poincaré extended the notion of Euler-Poincaré characteristic from
dimension 2 to the general case but proved the Poincaré-Hopf Theorem in dimension 2
only without extending it in the general dimension.
The meaning of Poincaré-Hopf Theorem is that Euler-Poincaré characteristic is a measure of the obstruction to constructing a continuous vector field tangent to the considered
manifold, without singularity.
One of the motivations of the Poincaré-Hopf Theorem is the study of differential
equations in terms of integral curves of an appropriate vector field. The singular points
of the vector field are points of equilibrium in dynamical systems.
That is the reason for which Poincaré-Hopf Theorem has many applications: mathematical economics, optimisation of communication systems, electrical engineering, applied
probability (cooperative dynamical systems), statistical complexity, particle physics (electromagnetic fields), structure of materials: stability of molecular complexes in chemistry,
crystallography, graphics applications, astrophysics: magnetic fields, etc... The interested
reader should experience to search for “Poincaré-Hopf Theorem” on his/her favorite web
search engine.
The first part of the chapter is devoted to various definitions of the index of a vector
field in an isolated singularity.
2.1
The index of a vector field.
In this section, one gives different ways to define the index of a vector field in an isolated
singular point. The obstruction theory definition will be useful for the following and
provides a geometrical meaning to characteristic classes. One can find generalisation of
the theory to non-isolated singularities in [BLSS2] (see [BSS] for a systematic study).
The index of a vector field at an isolated singularity can be defined in various ways.
We limit ourself to the “classical” ones (see [B3, BSS]).
In a first step, we consider vector fields in Euclidean space, then we will define the
index for vector fields tangent to a manifold.
Let Ω be an open subset in Rn with coordinates (x1 , . . . , xn ). Let
13
v=
n
X
fi ∂/∂xi
i=1
be a vector field on Ω. The vector field is said to be continuous, smooth, analytic,
according as its components {f1 , . . . , fn } are continuous, smooth, analytic, respectively.
A singularity a of v in Ω is a point where all of its components vanish, i.e., fi (a) = 0 for
all i = 1, . . . , n. The singularity is isolated if at every point x near a there is at least one
component of v which is not zero.
Let v be a continuous vector field on Ω with an isolated singularity at a, and let B(a)
be a small ball in Ω around a so that there is no other singularity of v within B(a). Let
us define the Gauss map
γ : ∂B(a) = S(a) ∼
= Sn−1 −→ Sn−1
by γ(x) = v(x)/kv(x)k.
Definition 2.1 The (local) index of v at a, denoted by I(v, a), is the degree of the Gauss
map γ : Sn−1 → Sn−1 .
The local index does not depend on the choice of the small ball B(a), on the choice
of coordinates nor on the choice of orientations.
Remark 2.2 In the following, we will use also a different kind of singularities for a
vector field, that M.-H. Schwartz called second type singularities. Let us introduce these
singularities.
Given a vector field v defined on the boundary S(a) of the ball B(a) of radius 1,
centred in a, there are many ways to extend the vector field inside B(a). Two are the
most natural. Let us denote by Sε (a) the sphere of radius ε, 0 < ε ≤ 1. If x ∈ S(a) the
vector v(εx) at the point εx ∈ Sε (a) is defined either as v(εx) = εv(x) or as v(εx) = v(x).
In the first case, the vector field v will be 0 in a, that is the already defined singularity
type. We will call it, according to M.-H. Schwartz, singularity of first type.
In the second case the extension is not defined at a (see [Sc4]), but it defines a cycle
κ(v) in the fibre Ta Rn of the tangent bundle to Rn . We will call it, again according to
M.-H. Schwartz, singularity of second type.
Whatever the type of singularity, the index I(v, a) of v at the isolated singularity a is
well defined by the Definition 2.1.
Proposition 2.3 Let us consider a second type singularity, then the index of the cycle
κ(v) in the punctured fibre Ta Rn \ {0} is equal to I(v, a).
Proof:
Let us denote by s0 the zero section of the tangent vector bundle T Rn . The
tangent bundle T Rn is trivial over B(a), as well as the bundle T × Rn = T Rn \ Ims0 (not
anymore a vector bundle). The fibre of T × Rn at a is Ta Rn \{0} ∼
= Rn \{0} and, restricted to
n
B(a), the bundle is homeomorphic to B(a)×(R \{0}). The vector field v defines a section
of T × Rn over S(a) whose image by the second projection B(a) × (Rn \ {0}) → Rn \ {0}
is equal to κ(v), by definition. One concludes by the Definition 2.1.
14
Let us consider now a n-dimensional smooth manifold M . A continuous vector field
on M is a section of its tangent bundle T M (see 3.1). Giving a local chart (Ua , φ) on M ,
where φ : Ua → Bn , a vector field on M is locally expressed as above:
Let us denote by xi = xi ◦ φ the coordinate functions of φ, i.e. the local coordinates
in Ua . We denote by ∂/∂xi the tangent vector at x defined by
∂
∂
(h ◦ φ−1 )|φ(x)
(h) =
∂xi
∂xi
for a C ∞ function h : M → R.
Definition 2.4 Let us denote by x = (x1 , . . . , xn ) the local coordinates of the manifold
M in the open neighborood Ua , a vector field v can be written in terms of the basis ∂/∂xi
of the tangent vector space Tx M
v=
n
X
fi
i=1
∂
.
∂xi
(2.5)
The functions (f1 , . . . , fn ) are called coordinates of the vector v in Ua . The vector field
is said to be continuous, smooth, analytic, according as its components {f1 , . . . , fn } are
continuous, smooth, analytic, respectively.
For simplicity, in the following, we will identify coordinates in Ua and Bn , omitting φ
and we will denote xi for xi .
A singularity (“first type singularity”) a of the vector field v is a point in which all
coordinate fi vanish. One can also define “second type singularities” of a vector field v
in the same way than in the Euclidean situation.
The index of the vector field v at a is well defined in both cases as the degree of the
Gauss map
γ : ∂φ−1 (Bn ) ∼
= Sn−1 −→ Sn−1
2.2
The index - Definition by obstruction theory
The index can be also defined in the following way: Let M be a differentiable manifold
of dimension n. The tangent bundle to M , denoted by T M , is a real vector bundle (see
section 3.1) of rank n, whose fibre in a point x of M is the tangent space to M at x,
denoted by Tx (M ) and is isomorphic to Rn . The vector bundle T M is locally trivial, i.e.
there is a covering of M by open subsets {Ua } such that the restriction of T M to each
Ua is homeomorphic to Ua × Rn .
Let us denote by s0 the zero section of T M , we will consider the bundle (not any more
a vector bundle) T × M = T M \ s0 (M ). Its fibre in a point x ∈ M is Tx× M ∼
= Rn \ {0}.
Let us consider a ball B(a) centred in a, contained in an open chart Ua over which
T M is trivial and sufficiently small so that a is the only singular point of v in B(a). One
can think of B(a) as an n-cell, in view of the generalisation we will perform later (3.15).
The vector field v defines a section of T M without zero over S(a) = ∂B(a), hence a map
pr2
v
S(a) ∼
= Sn−1 −→ T × M |Ua ∼
= Ua × (Rn \ {0}) −→ Rn \ {0}
15
(2.6)
Figure 3: Obstruction theory
where pr2 is the second projection.
One obtains a map
pr2 ◦v
Sn−1 ∼
= ∂B(a) −→ Rn \ {0}
hence an element λ(v, a) in πn−1 (Rn \ {0}). One knows that this homotopy group is Z.
The generator +1 can be interpreted in the following way:
Let us consider the radial vector field vrad , that is
vector field pointing outwards
Pthe
n
the ball B(a) along S(a), image of the vector field i=1 ∂/∂xi in Rn . Image of vrad in
Rn \ {0} is Sn−1 and the map pr2 ◦ vrad is the identity of Sn−1 .
Lemma 2.7 There is a homotopy
ψ : S(a) × [0, 1] → T × (B(a)|S(a) ) ⊂ T × (M |S(a) )
such that
∂Imψ = v(S(a)) − I(v, a) · pr2 ◦ vrad (S(a))
(2.8)
Proof:
Let us suppose without loss of generality that v is an unitary vector field on
×
S(a). For t 6= 0, let us define in the fibre Ttx
(M )
ψt (x) = ψ(t, x) = the unitary vector parallel to v(x) at the point tx.
If t goes to 0, then ψ0 (x) is the unit vector in T0× (M ) parallel to v(x) and with origin 0.
Therefore ψ0 (S(a)) is a cycle in the fiber T0× (M ) whose index is I(v, 0). In the case of the
radial vector field vrad on S(a), the cycle is the projection pr2 ◦ vrad (S(a)) over the fibre
at 0 (by local triviality of the bundle) and it has index I(vrad , 0) = 1. One concludes the
Lemma.
16
Proposition 2.9 The integer λ(v, a) equals to the index I(v, a) by the isomorphism
πn−1 (Rn \ {0}) ∼
= Z.
By classical homotopy theory, the map
pr2 ◦v
Sn−1 ∼
= ∂B(a) −→ Rn \ {0}
extends to a map
B(a) −→ Rn \ {0}
if and only if the element λ(v, a) is zero in πn−1 (Rn \ {0}), i.e. the vector field v extends
within the ball B(a) if and only if the index I(v, a) is zero.
∂B(a)∼
= Sn−1 −−→ Rn \ {0}


?%
y
B(a) ∼
= Bn
(2.10)
That construction is the basis of obstruction theory, it will be generalised in chapter 3.
Remark 2.11 Let v be a vector field defined in a neighborhood U of a ∈ Rn and let B
be a n-ball containing a such that a is an isolated singularity of v in B. Then, the index
of v is determined by the behavior of v on the boundary ∂B of the ball, independently of
what happens inside the ball.
2.3
Relation with the Gauss map
Let N a compact k-manifold with boundary in Rk . The Gauss map
g : ∂N → Sk−1
assigns to each x ∈ ∂N the outward unit normal vector at x. The degree of the Gauss
map is well defined as the class of g(∂N ) in Hk−1 (Sk−1 ) ∼
= Z.
Lemma 2.12 (Hopf ) ([Mi1], §6, Lemma 3) If v is a smooth vector field on N with
isolated P
singularities ai and v points outward of N along the boundary, then the sum of
indices
I(v, ai ) equals the degree of the Gauss mapping from ∂N to Sk−1 .
Proof:
For each singular point ai one considers a small (closed) ball B(ai ) with
center ai and
S which do not intersect each other. The vector field v has no singularity in
W = N \ i B(ai ).
Let us consider a compact n-manifold without boundary M ⊂ Rk . Let Nε denote the
closed ε-neighbourhood of M (i.e. the set of all x ∈ Rk with kx − yk < ε for some y ∈ M ).
For ε sufficiently small, Nε is a smooth manifold with boundary.
Theorem 2.13 Let v be a vector field with isolated
singularities ai , on a compact manifold
P
k
without boundary M ⊂ R , the index sum
I(v, ai ) is equal to the degree of the Gauss
mapping
g : ∂Nε → Sk−1 ,
where Nε denotes the closed ε-neighbourhood of M in Rk .
17
We reproduce the proof due to Milnor [Mi1], §6, Theorem 1. That proof has been
delivered in December 1963 in lectures in University of Virginia. The procedure used by
Milnor is the same as the one developed independently and at the same time by M.-H.
Schwartz [Sc1], in her definition of radial extension in the framework of stratified singular
varieties. More precisely, the idea is to extend a vector field v defined on the manifold
M with index I(v, a; M ) at the isolated singularity a, as a vector field w in the ambient
space Rk that has also an isolated singularity at a with the same index I(w, a; Rk ) =
I(v, a; M ). The principle is to sum the parallel extension of v in a neighbourhood of a
with a transversal vector field.
Proof:
For x ∈ Nε , let r(x) be the closest point of M . The vector x − r(x) is
perpendicular to the tangent space of M at r(x), for otherwise, r(x) would not be the
closest point of M . If ε is sufficiently small, then the restriction r(x) is smooth and well
defined.
We consider the squared distance function (for the Euclidean metric in Rk ):
φ(x) = kx − r(x)k2
whose gradient vector field is
gradφ(x) = 2(x − r(x)).
On one hand, the gradient vector field is a vector field defined in Nε that is zero along M ,
that is transverse to ∂Nε going outward and that increases with the distance to M . For
each point x at the level surface ∂Nε = φ−1 (ε2 ), the outward unit normal vector, called
transversal vector, is given by
g(x) = gradφ(x)/kgradφ(x)k = (x − r(x))/ε.
On the other hand, in each point x ∈ Nε , the vector v1 (x) = v(r(x)) is a parallel
extension of v.
Extend v to a vector field w on the neighbourhood Nε by setting
w(x) = (x − r(x)) + v1 (x).
The vector field w points outward along the boundary ∂Nε , since the inner product
w(x) · g(x) is equal to ε > 0. In fact w vanish only at the zeros of v in M . That is clear
because the two summands (x − r(x)) and v1 (x) are orthogonal.
Now, the index of w at the zero a, computed in Rk is equal to P
the index of v at a,
computed in M and, according to the Lemma 2.12, the index sum
I(v, a) is equal to
the degree of g which proves the theorem.
P
The Theorem is another way to see that if M is compact, the sum
I(v, ai ) for all
singularities of v does not depend on v. We will see (Theorem 5.12) that, with suitable
vector fields, the result extends to the case of singular variety M .
2.4
Poincaré-Hopf Theorem
There are many ways to prove Poincaré-Hopf Theorem. They correspond to the different
viewpoints and definitions of the index. The interested reader can consult [Li], [Mi1]
(Hopf and Gauss map), [GP] (Lefschetz fix points theory), [Hirs] (Intersection numbers).
18
2.4.1
The smooth case without boundary
Theorem 2.14 [Poincaré-Hopf Theorem] Let M be a compact differentiable manifold,
and let v be a continuous vector field on M with finitely isolated singularities ai . One has
X
χ(M ) =
I(v, ai )
i
Proof: Firstly we prove the Theorem in the orientable case, then in the non-orientable
case. We will follow the Milnor proof which is close to the generalisation to singular
varieties that we will provide in the next chapters.
1) Orientable case.
The idea of the proof is the following: In a first step, one shows that the sum of
indices of a continuous tangent vector field with isolated singularities does not depend
of the choice of the vector field. The second step of the proof consists in describing a
particular vector field for which the sum of indices is equal to χ(M ).
For the first step, Theorem 2.13 provides directly the result.
For the second step, such a vector field is given for instance by the gradient field
associated to a Morse function. Another possibility is to consider the Hopf vector field
H of which we recall the construction (see [Ste], p. 202). Let us consider a triangulation
K of M and a barycentric subdivision K 0 of K. The Hopf vector field will be tangent to
simplexes of K 0 , with a singularity in every vertex of K 0 , i.e. in every barycenter of K.
On every simplex [σ̂, τ̂ ] of K 0 , where σ̂ and τ̂ are barycenters of σ and τ , with σ < τ , the
vector field H is going in the direction from σ̂ to τ̂ . For example it is going outward all
vertices of K. One complete with a vector field whose integral curves are given (in the
2-simplex) in Figure 4. The higher dimensional case is easy to understand.
Figure 4: Hopf vector field
The Hopf vector field H has a singularity of index (−1)i inP
the barycenter of every
i-simplex of K. The sum of indices of H in all singularities is ni=0 (−1)i ki where ki is
the number of i-dimensional simplexes of K, so it is equal to χ(M ).
19
2) The non-orientable case:
f → M . On one hand, if v is a
Let us consider the oriented double covering π : M
continuous vector field on M with isolated singular points ai of index I(v; ai ), then on
f with isolated singular
can define a lifting ṽ of v which is a continuous vector field on M
j
j
points ai , j = 1, 2 such that π(ai ) = ai . P
As π is a local homeomorphism,
one has
P
j
j
I(v; ai ) = I(v; ai ) for j = 1, 2. One obtains i,j I(v; ai ) = 2 i I(v; ai ). On the other
f) = 2χ(M ) (use suitable triangulations). One conclude the Poincaréhand, one has χ(M
Hopf Theorem :
f) = 1/2
χ(M ) = 1/2 · χ(M
X
I(v; aji ) =
i,j
2.4.2
X
I(v; ai ).
i
Consequences of Poincaré-Hopf Theorem
As an important consequence of the Poincaré-Hopf Theorem, one has the following
Corollary 2.15 Let M be a compact smooth manifold, if χ(M ) 6= 0, then any continuous
vector field tangent to the manifold M admits at least a singular point. Reciprocally, every
compact manifold such that χ(M ) = 0 admits a continuous tangent vector field without
singularities.
The unitary sphere Sn with odd n satisfies χ(Sn ) = 0 and admits continuous tangent
vector fields without singularities. If n is even, χ(Sn ) = 2 and in that case every continuous
vector field tangent to Sn admits at least one singularity.
Corollary 2.16 Every compact odd dimensional manifold admits a continuous tangent
vector field without singularity.
The torus and the Klein bottle are the only one compact 2-dimensional surface admitting a non-zero continuous tangent vector field.
Lemma 2.17 For even-dimensional hypersurfaces, the Euler-Poincaré characteristic χ(M )
equals twice the degree of the Gauss map γ.
Proof:
Take the projection π : Sn → RPn and a regular value p ∈ RPn of the
composed map π ◦ γ : M → RPn . Take a differentiable vector field w on Sn with
isolated singularities in {a, b} = π −1 (p) of indices +1. The vector field v on M such that
v(x) = w(γ(x)) has a finite number of isolated singularities
{a1 , . . . , arP
} = γ −1 (a) and
P
r
{b1 , . . . , bs } = γ −1 (b). One
deg(γ) = i=1 I(v; ai ) = sj=1 I(v; bj ), on
Prone hand, one
Phas
the other hand χ(M ) = i=1 I(v; ai ) + sj=1 I(v; bj ). That gives the Lemma.
20
2.4.3
The smooth case with boundary
Let M be an oriented manifold with boundary, one has a similar theorem:
Theorem 2.18 [Poincaré-Hopf Theorem with boundary] Let M be a compact manifold
with boundary ∂M embedded in an oriented differentiable manifold N . Let v be a nonsingular continuous vector field tangent to N , strictly pointing outwards (resp. inwards)
M along the boundary ∂M . Then:
1. v can be extended to the interior of M as a vector field tangent to M with finitely
many isolated singularities ai .
2. The total index of v in M is independent of the way we extend it to the interior of
M . In other words, the total index of v is fully determined by its behaviour near the
boundary.
3. If v is everywhere transverse to the boundary and pointing outwards from M , then
one has
X
χ(M ) =
I(v, ai ).
(2.19)
i
If v is everywhere transverse to ∂M and pointing inwards then
X
χ(intM ) = χ(M ) − χ(∂M ) =
I(v, ai ).
(2.20)
i
Proof: The first statement is proved by obstruction theory (section 2.2). The vector
field can be extended without singularities to the (n − 1)-skeleton of M . Then we extend
it to the n-cells introducing (if necessary) a singular point for each n-cell.
The second statement is also a general result in obstruction theory, that can be obtained (for instance) as a consequence of statement 3 (see also 3.3 and [Ste]).
A proof of the third statement goes in the following way: Like in Theorem 2.13,
on consider the closed ε-neighbourhood of M , denoted by Nε . If the vector field is
pointing outward along ∂M , then it can be extended over the neighbourhood Nε so that
the extended one points outward along ∂Nε . The extension w is defined as before by
w(x) = (x − r(x)) + v(r(x)) and is a continuous vector field near ∂M . In this case, Nε is
not necessarily of class C ∞ , but only a C 1 -manifold. Nevertheless, the same argument as
in the case “without boundary” can be carried out (see [Mi1] §6), that gives (2.19).
If the vector field is pointing inward along ∂M , one can extend v inside M with finitely
many isolated singularities ai of index I(v, ai ).
One proceeds to the following construction: the boundary ∂M admits a neighbourhood
∂M × [0, 1] in M and one can extend this neighbourhood as ∂M × [0, 2]. Let us call M 0
the new manifold M ∪ (∂M × [0, 2]). One has χ(M 0 ) = χ(M ) and ∂M 0 ∼
= ∂M . Let us
call C the collar ∂M × [1, 2]. One has χ(C) = χ(∂M ).
At the level C1 = ∂M ×{1}, one has the vector field v pointing inward M and outward
C. At the level C2 = ∂M × {2}, one considers any vector field v 0 pointing outward M 0
along ∂M 0 . Let us call w the vector field defined on ∂C which is equal to v and v 0 on
C1 and C2 respectively. The vector field w is defined on the boundary of C and pointing
21
outward C along the boundary. By (2.19) on C, one can extend w inside C with finitely
many isolated singularities bj and one has
X
χ(C) = χ(∂M ) =
I(w, bj ).
j
On M 0 one consider the vector field v 0 , which is v on M and w on C. It has isolated
singularities ai and bj and it is pointing outward M 0 . Again one can apply (2.19) (on M 0 )
and one has
X
X
X
χ(M ) = χ(M 0 ) =
I(v, ai ) +
I(w, bj ) =
I(v, ai ) + χ(∂M )
i
i
and the result.
Corollary 2.21 Let us suppose M is odd-dimensional, then
χ(∂M ) = 2 · χ(M )
Proof:
Let us denote by v a vector field pointing outwards D along the boundary,
like in 2.19 and let us consider the vector field w = −v. Then, w has same singularities
than v and, as M is odd-dimensional, in each singularity ai , one has I(w, ai ) = −I(v, ai ).
Equations 2.19 and 2.20 provide the result.
Corollary 2.22 Let us denote by M ⊂ Rk a compact manifold without boundary in an
odd-dimensional Euclidean space and by Nε the closed ε-neighbourhood of M in Rk . Then
one has
χ(∂Nε ) = 2 · χ(M )
22
3
Characteristic classes : the smooth case
In 1935 and independently, Stiefel, who was student of Hopf, and Whitney defined characteristic classes in cohomology for real manifolds. Stiefel considers the obstruction point
of view (for the construction of r-frames tangent to the manifold), computing homotopy
groups of so-called Stiefel manifolds. Whitney considers sphere bundles on a manifold M
and defines cohomology classes with coefficients in Z/2 = Z/2Z. The Stiefel and Whitney
methods are similar and represent the basis of obstruction theory. We call Stiefel-Whitney
classes of a vector bundle or of the associated sphere bundle, the classes obtained in that
way.
In 1942 Pontrjagyn defined classes for Grassmannian manifolds, using a decomposition
of these manifolds in terms of Schubert varieties, due to Ehresmann.
In his fundamental 1946 paper [Ch], Chern gave several constructions of characteristic
classes for Hermitian Manifolds. The paper provides basement for the relationship between obstruction theory, Schubert varieties, differential forms, transgression, etc... We
will briefly recall some of these definitions, either because they extend to the case of
singular varieties, or because they will be useful for the following.
Contribution of Wu Wen Tsün in the history of characteristic classes is important.
Among results, he proved the product formula for Stiefel-Whitney and Chern classes, he
gave a simple formulation of the decomposition of the Grassmann manifold of oriented
vector subspaces and he extended the definition of Chern classes for any complex vector
space on any finite simplicial complex.
As it happens often in Mathematics, one object, here the Chern classes, has (at least)
two definitions: the geometric definition allows to understand the signification of classes,
but it is difficult to proceed to effective computations in this context. The axiomatic
definition provides easy ways to compute effectively the classes but is less suitable to
understanding the origin and the meaning of the classes (see [MS, Hu]).
A trivial bundle is induced from a map to a point, so all its characteristic classes (except
the zero dimensional one) should be zero. More generally, equality of all characteristic
classes of two bundles is a necessary (and in some circumstances sufficient) test for their
equivalence. That is one of the important uses of characteristic classes.
The interested reader will find all wished references in the Dieudonné book [Di], 3,IV.
3.1
Fibre bundles.
In this section, we will denote by K either the real field R or the complex field C. We
provide elementary definitions and properties of vector and fibre bundles, as well as a
23
series of examples in the real and complex situations. The reader will find in the literature
suitable references for more definitions and properties (see for instance [Hirz] and [Hu]).
3.1.1
Vector bundles
Definition 3.1 A vector bundle E, over the field K, with base X and rank n is a topological space E, with a continuous map π : E → X, the projection, such that for every
point x ∈ X, the fibre Ex = π −1 (x) is a vector space of rank n over K.
A vector bundle satisfies the local triviality condition: for every point x ∈ X, there is
an open neighbourhood Ux in X and a homeomorphism φ : π −1 (Ux ) → Ux × Kn which
induces for every y ∈ Ux an isomorphism π −1 (y) → Kn .
A trivial bundle is a bundle for which one has “global” triviality, i.e. one can take
Ux = X in the previous condition.
Given a vector bundle E over X, one define in a natural way the dual vector bundle
∗
E and the bundle of k-vectors Λk E whose fibres are respectively (Kn )∗ and Λk Kn .
Vector bundles are special cases of fibre bundles that we recall now.
3.1.2
Fibre bundles
Let F a topological space and G a topological group which acts effectively and continuously on F . That means there is a continuous map G × F → F such that one has
g1 · (g2 · a) = (g1 g2 ) · a for g1 , g2 ∈ G and a ∈ F , and e · a = a if e is the identity element
in G. The action is effective means that if g · a = a for some a ∈ F then g = e.
Definition 3.2 A topological space E with a continuous projection π : E → X, is called
a fibre bundle with fibre F and structure group G if G acts effectively and continuously
on F and there are a system of coordinates (Ui , φi ) on X and continuous functions gij :
Ui ∩ Uj → G such that:
• {Ui } is an open covering of X and φi : π −1 (Ui ) → Ui × F is a homeomorphism
identifying π −1 (x) with the fibre {x} × F ,
• (φi ◦ φ−1
j )(x, a) = (x, gij (x) · a) for all x ∈ Ui ∩ Uj and a ∈ F .
The fonctions gij are called transition functions. They satisfy
gij ◦ gjk ◦ gki = id
for all i, j, k,
hence, they define a cocycle in Z 1 (X, G), then an element in H 1 (X, G). It is well known
that isomorphism classes of fibre bundles over X with fibre F and structural group G are in
a one-one correspondence with the elements of H 1 (X, G). The trivial bundle corresponds
to the element 1 ∈ H 1 (X, G). Fibre bundles in the same isomorphism class ξ ∈ H 1 (X, G)
are said associated bundles.
The fibre bundle is said differentiable if X is a differentiable manifold and G a real Lie
group, the gij being differentiable functions. The fibre bundle is said complex analytic if
X is a complex manifold and G a complex Lie group, the gij being holomorphic functions.
A section of the fibre bundle E is a continuous application s : X → E such that, for
every point x ∈ X, one has s(x) ∈ Ex = π −1 (x).
24
3.1.3
Examples of fibre bundles - real case
In order to provide examples of real vector bundles and fibre bundles, we will use the
following spaces:
The real projective space RPn is the space of lines through the origin of Rn+1 . The
Grassman manifold Gr (Rn ) is the space of all vector subspaces of dimension r of Rn .
Let R∞ be the vector space of all infinite sequences (x1 , x2 , . . .) whose elements xi are
real numbers, a finite number of them being nonzero. The infinite Grassmannian manifold
Gr (R∞ ) is the set of all r-dimensional subspaces in R∞ , i.e. the direct limit of the natural
sequence of inclusions
Gr (Rr ) ⊂ Gr (Rr+1 ) ⊂ Gr (Rr+2 ) ⊂ · · ·
We consider on Gr (R∞ ) the topology for which closed subsets are those whose intersections
with all Gr (Rr+k ) are closed.
Examples of real vector bundles are given by:
1. the tangent bundle T M to a differentiable manifold M . That is the set of all pairs
(x, v) such that x ∈ M and v is a vector tangent to M at the point x, i.e. an
element of Tx M . If M is an n-manifold, then T M is a real vector bundle with rank
n over M , the fibre is Rn .
In particular, one has the bundle T Sn tangent to the sphere Sn , that is a trivial
bundle if n = 1, a non trivial bundle if n = 2. One has also the bundle T RPn
tangent to RPn .
2. the normal bundle to a differentiable n-manifold M embedded in Rn+k . That is the
set of all pairs (x, v) ∈ M × Rn+k such that v is orthogonal to the tangent space
Tx M ∼
= Rn in Tx (Rn+k ) ∼
= Rn+k .
3. the canonical bundle over RPn also called tautological bundle and denoted by γ1n :
γ1n → RPn
(3.3)
This line bundle is the set of all pairs {(λ, v)} where λ is an element of RPn , i.e. a
line passing through the origin of Rn+1 and v a vector of λ. The canonical bundle
is not trivial, and this fact is the basis for the axiomatic definition of characteristic
classes.
4. the canonical bundle γrn over the Grassman manifold Gr (Rn ). That is the set of all
pairs {(P, v)} where P is an element of Gr (Rn ) and v a vector in P . One has the
bundle projection
γrn → Gr (Rn )
and γrn is a vector bundle with rank r.
The bundle is also called universal bundle for vector bundles of rank r. That means
that every bundle ξ with rank r over a (paracompact) topological space X is isomorphic to f ∗ (γrn ) for some f : X → Gr (Rn ) with sufficiently large n.
25
5. the universal bundle
γr → Gr (R∞ )
set of all pairs {(P, v)} where P is an element of Gr (R∞ ) and v a vector of P . It is
universal for all rank r-vector bundles.
In the case r = 1, that is the bundle γ1 → RP∞ .
6. the Stiefel manifold, denoted by Vr (Rn ) is the set of r-frames in Rn , that is the set
of ordered r-uples (v1 , . . . , vr ) of r linearly independent vectors in Rn . (see Steenrod
0
).
[Ste] where this manifold is denoted by Vr,n
One has a homotopy Vr (Rn ) ∼
= Vr,n = O(n)/O(n − r).
The natural map Vr,n → Gr (Rn ) is a principal fibre bundle, i.e. the fibre O(r)
coincides with the structural group. That is an universal bundle for fibre bundles
whose basis has dimension ≤ n − r − 1.
The vector bundle γrn → Gr (Rn ) is a bundle associated to Vr,n → Gr (Rn ) with fibre
Rr .
7. the bundle Vr (T M ) of r-frames tangent to a n-differentiable manifold M , i.e. the
set of all pairs (x, (v1 , . . . , vr )) where x is a point of M and (v1 , . . . , vr ) is a r-frame
in the fibre Tx M over x. That is the fibre bundle over M whose fibre at x is the
manifold Vr (Tx M ) of all r-frames in Tx M . The fibre is the Stiefel manifold Vr (Rn ).
Note that a section of this bundle in Stiefel manifolds is a r-uple of linearly independent sections of the vector bundle T M .
3.1.4
Examples of fibre bundles - complex case
One considers the complex projective space CPn whose homogeneous coordinates will be
denoted by (x0 : x1 : . . . : xn ). The projective space is covered by open subsets {Ui }i=0,...n
homeomorphic to Cn and whose coordinates are (x0 , x1 , . . . , xi−1 , 1, xi+1 , . . . , xn ).
We will consider the complex Grassmannian manifolds Gr (Cn ) and Gr (C∞ ) in a similar
way than in the real case.
1. the complex tangent bundle T M to a complex n-dimensional manifold M is a fibre
bundle over M . Each fibre Tx M has a complex structure and is isomorphic to Cn .
In particular, one has the tangent bundle T CPn to CPn .
2. the canonical bundle γ1n over CPn . also called tautological or universal bundle and
denoted by O(−1) in algebraic geometry:
γ1n → CPn
(3.4)
This line bundle is the set of all pairs {(λ, v)} where λ is an element of CPn , i.e.
a complex line passing through the origin of Cn+1 and v a vector in λ. That is the
fibre over λ is the line λ.
γ1n = {(λ, v) ∈ CPn × Cn+1 |v ∈ λ}.
26
With the previous homogeneous coordinates in CPn , the transition functions of γ1n
in Ui ∩ Uj are defined by xi /xj .
3. the “hyperplane” bundle H over CPn , dual of the canonical bundle. It is denoted
by O(1) in algebraic geometry. With the previous homogeneous coordinates, the
transition functions of the hyperplane bundle in Ui ∩ Uj are defined by (xi /xj )−1 .
The hyperplane x0 = 0 with the induced orientation, is CPn−1 , that is a 2(n − 1)cycle in H2(n−1) (CPn ; Z). The Poincaré dual cohomology class hn is a generator of
H 2 (CPn ; Z). We will see that c(H) = 1 + hn .
4. the universal bundle
γrn → Gr (Cn )
is the set of all pairs {(P, v)} where P is an element of Gr (Cn ) and v a vector of P .
That is a vector bundle of rank r over Gr (Cn ).
Every complex vector bundle ξ with rank r over a (paracompact) topological space
X is isomorphic to f ∗ (γrn ) for some f : X → Gr (Cn ) with sufficiently large n.
5. the universal bundle
γr → Gr (C∞ )
is the set of all pairs {(P, v)} where P is an element of Gr (C∞ ) and v a vector of
P . In particular, one has the bundle γ1 → CP∞ .
The bundle γr is universal for all rank r-vector bundles.
6. One defines the Stiefel manifold Vr (Cn ) which is the set of r-frames in Cn , that is
the set of ordered r-uples (v1 , . . . , vr ) of C-linearly independent vectors in Cn (see
0
).
Steenrod [Ste] where the Stiefel manifold is denoted by Wr,n
One has a homotopy
Vr (Cn ) ∼
= Wr,n = U (n)/U (n − r).
The fibre bundle Wr,n → Gr (Cn ) is a principal bundle with fibre and structural
group U (r).
The fibre bundle Wr,n → Gr (Cn ) is an universal bundle for bundles which basis has
dimension ≤ 2(n − r).
The vector bundle γrn → Gr (Cn ) is a bundle associated to Wr,n → Gr (Cn ) with fibre
Cr .
7. One define the bundle of complex r-frames tangent to the complex n-manifold M ,
i.e. the set of all pairs (x, (v1 , . . . , vr )) where x is a point of M and (v1 , . . . , vr ) is
a r-frame in the fibre Tx M over x. That is the fibre bundle whose fibre at x is the
manifold Vr (Tx M ) consisting of all complex r-frames in Tx M . The “typical” fibre
is the Stiefel manifold Vr (Cn ).
Note that a section of this bundle in complex Stiefel manifolds is a r-uple of Clinearly independent sections of the complex vector bundle T M .
27
3.2
General obstruction theory
Let us recall the idea of the construction of characteristic classes by obstruction theory,
following Steenrod [Ste], part III.
We have seen that the meaning of Poincaré-Hopf Theorem is that the Euler-Poincaré
characteristic of a manifold M is a measure of the obstruction for the construction of a
vector field tangent to M . In a more general way, the aim of the obstruction theory is
to define invariants providing a measure of the obstruction to the construction of linearly
independent sections of vector bundles. In a more precise way, the objective is to answer
to questions of the following type:
Let E be a vector bundle of rank n on a variety X and fix r such that 1 ≤ r ≤ n, is
it possible to construct r sections of E, linearly independent everywhere?
It is obviously possible to define such sections on the 0-skeleton of a triangulation of
X. So, the question becomes the following:
Let us consider a triangulation of X. Performing the construction of r independent
sections by increasing dimension of the simplexes, up to what dimension can we proceed?
Arriving to this obstruction dimension, is it possible to evaluate the obstruction?
At that point, let us make a comment: Classical obstruction theory uses a triangulation
of the considered space. In the following we will use a slightly different viewpoint, taking
into account the fact that we want to deal with the singular case. It appears that in the
singular case, the good decomposition to be taken into account for the construction of the
sections is not a triangulation of the space but a dual cell decomposition in the ambient
space. That is the reason for which, we will work on a cell decomposition, already in the
non-singular case.
Let us consider a (simplicial or cellular) complex K and a subcomplex L. We will
denote by X = |K| and Y = |L| the respective geometric realisations. The q-skeleton of
K is denoted by K q , that is the subcomplex consisting of all simplexes (or cells) whose
dimension is less or equal to q. Let us denote Xq = |K q | the associated space.
We consider a fibre bundle E with basis X and fibre F . To consider a section defined
in a trivialisation open subset for the bundle provides a map with target F , as we already
seen, see for instance (3.16).
Aim of obstruction theory is to describe the problem of extension of maps f : Y → F
to all of X, by successive extensions of the map from Xq to Xq+1 . Let us suppose that
the function f : X → F is already known on Xp−1 and let us denote it by fp−1 . Let dp
an oriented p-cell, fp−1 is well defined on the boundary ∂dp and determines an element
[fp−1 |∂dp ] ∈ πp−1 (F ).
Definition 3.5 The relative cochain denoted by c(fp−1 ) ∈ C p (K, L; πp−1 (F )) and defined
by
c(fp−1 )(dp ) = [fp−1 |∂dp ] ∈ πp−1 (F )
(3.6)
is called obstruction cochain (for the extension of fp−1 to Xp ).
The function fp−1 can be extended to Xp if and only if c(fp−1 ) = 0. In particular,
if πi (F ) = 0 for i = 1, . . . , j − 1, then every function fY : Y → F can be extended to
f j : Xj → F .
28
Lemma 3.7 If fp−1 is homotopic to gp−1 , then c(fp−1 ) = c(gp−1 ).
Proof:
In fact, as fp−1 |∂dp ∼
= gp−1 |∂dp , one has [fp−1 |∂dp ] = [gp−1 |∂dp ].
Theorem 3.8 c(fp−1 ) is a cocycle.
Proof:
Let τ p+1 a (p + 1)-cell. One has to show that δ[c(fp−1 )](τ p+1 ) = 0. One has
X
δ[c(fp−1 )](τ p+1 ) = c(fp−1 )[∂τ p+1 ] = c(fp−1 )( [τ p+1 : dpi ]dpi ) =
X
[τ p+1 : dpi ]c(fp−1 )(dpi ) =
X
[τ p+1 : dpi ][fp−1 |∂dpi ]
where the sum is taken on all cells dpi which are faces of τ p+1 . Let us suppose thatP
incidence
of all faces dpi of τ p+1 with τ p+1 is positive, then denoting fp−1 |∂dpi = αi , one has αi = 0,
and the result. If incidence is not positive, then [τ p+1 : dpi ][fp−1 |∂dpi ] = αi is the element
of πp−1 (F ) obtained from the function fp−1 restricted
to the boundary ofPthe face dpi with
P
the orientation induced from τ p+1 and one has [τ p+1 : dpi ][fp−1 |∂dpi ] = αi = 0.
3.2.1
The difference cochain
Let fp−1 and gp−1 two extensions on Xp−1 of the same fp−2 : Xp−2 → F . We intend to
provide a “measure” of their difference. Let dp−1 a (p − 1)-cell in (K). The two functions
fp−1 |dp−1 and gp−1 |dp−1 coincide on the boundary ∂dp−1 . The cell dp−1 is homeomorphic
p−1
p−1
both to the north hemisphere D+
, and to the south hemisphere D−
, of the sphere
p−1
p−1
Sp−1 . One can interpret fp−1 |dp−1 , resp. gp−1 |dp−1 , as a function of D+
, resp. D−
in F .
p−2
p−1
These functions coincide on the equator S , homeomorphic to ∂d , hence they define
a function γ : Sp−1 → F . Annulation of the homotopy class [γ] ∈ πp−1 (F ) is a necessary
and sufficient condition to deform gp−1 |dp−1 in fp−1 |dp−1 .
Definition 3.9 The difference cochain d(fp−1 , gp−1 ) ∈ C p−1 (K, L; πp−1 (F )) is defined by
d(fp−1 , gp−1 )(dp−1 ) = (−1)p [γ] ∈ πp−1 (F ).
The difference cochain is a relative cochain of K modulo L. It vanishes if and only if
fp−1 ∼
= gp−1 relatively to Xp−2 . If hp−1 is a third extension of fp−2 then one has
d(fp−1 , hp−1 ) = d(fp−1 , gp−1 ) + d(gp−1 , hp−1 ).
If fp−1 is an extension of fp−2 and cp−1 ∈ C p−1 (K, L; πp−1 (F )) is a relative cochain, then
there is an extension gp−1 of fp−2 such that d(fp−1 , gp−1 ) = cp−1 .
Theorem 3.10 One has
δd(fp−1 , gp−1 ) = c(fp−1 ) − c(gp−1 ).
That means that the difference of the obstruction cocycles of two extensions of fp−2
is a coboundary.
29
Lemma 3.11 If fp−2 can be extended as a function fp−1 : Xp−1 → F , then all obstruction
cocycles c(fp−1 ) for extension of fp−2 to Xp−1 belong to the same cohomology class
c̄(f ) ∈ H p (K, L; πp−1 (F )).
Theorem 3.12 Let fp−1 : Xp−1 → F , then fp−2 extends to fp : Xp → F if and only if
c̄(f ) = 0.
3.2.2
The obstruction class
Here we are using cohomology with local coefficients H p (K; {πp−1 (F )}), i.e. bundle of
abelian groups which associate to each point x of X the coefficient group πp−1 (Fx ).
Let us suppose that πi (F ) = 0 for i ≤ p − 2. Then one can construct a fonction fi
without singularity for 1 ≤ i ≤ p − 1.
Definition 3.13 Let us suppose that πi (F ) = 0 for i ≤ p − 2. The primary obstruction
class is the class of the obstruction cocycle [c(fp−1 )], that is
c̄(f ) ∈ H p (K, L; πp−1 (F )).
Let us remark that in general, the system of coefficients {πp−1 (F )} is twisted.
3.3
Case of the tangent bundle
We study the particular case of the (real) tangent bundle to a differentiable smooth
manifold or the (complex) tangent bundle to an analytic complex manifold. We will
denote by K the field R or C, according to the situation.
Let M be a manifold of dimension n, over K, endowed with an euclidean (or hermitian)
metric. The tangent bundle to M , denoted by T M , is a vector bundle of rank n over K,
whose fibre in a point x of M is the tangent vector space to M in x, denoted by Tx (M )
and is isomorphic to Kn . The vector bundle T M is locally trivial, i.e. there is a covering
of M by open subsets such that the restriction of T M to U is isomorphic to U × Kn .
The objective is to evaluate the obstruction to the construction of r sections of T M
linearly independent (over K) in each point, i.e. an r-frame:
Definition 3.14 An r-field on a subset A of M is a set v (r) = {v1 , . . . , vr } of r continuous
vector fields tangent to M , defined on A. A singular point of v (r) is a point where the
vectors (vi ) fail to be linearly independent. A non-singular r-field is called an r-frame.
The r-frames are sections of the fibre bundle Vr (T M ) over M . That is the fibre bundle
associated to T M and whose fibre at the point x of M is the set of r-frames of Tx (M ).
The fibre is the Stiefel manifold denoted by Vr (Kn ) that we described in 3.1.3 in the real
case and in 3.1.4 in the complex case.
To construct r linearly independent sections of T M over a subset A of M is equivalent
to construct a section of Vr (T M ) over A.
Let us consider the following situation: (K) is a cell decomposition of M sufficiently
small so that every cell d is included in an open subset U over which Vr (T M ) is trivial.
30
One remarks that trivialisation open sets for Vr (T M ) are the same that the ones of T (M ).
There exists always such a cell decomposition.
Let us consider the following question:
Let us suppose that one has a section v (r) of Vr (T M ) on the boundary ∂d of the k-cell
d. Is it possible to extend this section in the interior of d ? Is the answer is no, can one
evaluate the obstruction for such an extension ?
In order to answer the question, we need to define the notion of index of an r-field in a
singular point and we need some notions and results on general obstruction theory. That
is aim of the following sections. Then we will apply these results to the real and complex
case, that is to define Stiefel-Whitney and Chern classes.
3.3.1
Index of a r-frame
Let us consider an r-field v (r) defined on the boundary ∂d of a k-cell d of the cell decomposition (D) of M . In the same way than in 2.2, v (r) is a section of the bundle Vr (T M ),
defined on the boundary of d. It provides a map
(r)
pr2
v
∂d −→ Vr (T M )|U ∼
= U × Vr (Kn ) −→ Vr (Kn ),
(3.15)
where pr2 is the second projection.
Figure 5: Obstruction theory
One obtains a map
(r)
pr2 ◦v
Sk−1 ∼
= ∂d −→ Vr (Kn )
(3.16)
which defines an element of πk−1 (Vr (Kn )) denoted by [ξ(v (r) , d)].
Let us suppose that [ξ(v (r) , d)] = 0, then, by classical homotopy theory, the map
k−1
S
→ Vr (Kn ) defined on the boundary Sk−1 of the ball Bk can be extended inside the
31
ball. In another words, if [ξ(v (r) , d)] = 0, then the map ∂d → Vr (Kn ), i.e. the r-frame,
can be extended inside the cell d. One can resume the situation by the following diagram
similar to (2.10).
∂d ∼
=Sk−1 −−→ Vr (Kn )


?%
y
d∼
= Bk
This means that there is no obstruction to the extension of the section v (r) inside d. This
happens for example in the case πk−1 (Vr (Kn )) = 0.
In order to answer to the previous question, we need to know the homotopy groups
of Vr (Kn ). The homotopy groups πk−1 (Vr (Kn )) have been computed by Stiefel and by
Whitney (see [Sti]) in the cases K = R and C. One has the following result:
Let Vr (Rn ) be the Stiefel manifold of r-frames in Rn , one has:


0
n
πi (Vr (R )) = Z


Z2
for i < n − r
for i = n − r even or i = n − 1 if r = 1
for i = n − r odd and r > 1
For the Stiefel manifold Vr (Cn ) of (complex) r-frames in Cn , one has:
(
0 for i < 2n − 2r + 1
n
πi (Vr (C )) =
Z for i = 2n − 2r + 1
(3.17)
(3.18)
A generator of the first non-zero homotopy group can be described in the following
way. Let us give it in the real framework, then in the complex one.
In the real case, let us denote p = n − r + 1, one describes a generator of πn−r (Vr (Rn )).
Let us fix a (r − 1)-frame in Rn . It defines a (r − 1)-subspace of Rn whose complementary
is a real space Rp . The unit sphere in Rp denoted by Sp−1 is oriented. Let us consider,
for each point w of the sphere, a r-frame consisting of the vector w and the fixed (r − 1)frame, one obtains an element of Vr (Rn ). The induced map from the oriented sphere Sp−1
to Vr (Rn ) defines a generator of πp−1 (Vr (Rn )).
In the complex case, let us denote 2p = 2(n − r + 1), one describes a generator of
π2p−1 (Vr (Cn )) ∼
= Z. Let us fix a (r − 1)-frame in Cn . It defines a (r − 1)-subspace of Cn
whose complementary is a complex space Cp . The unit sphere in Cp denoted by S2p−1 is
oriented, the orientation being induced by the natural one of Cp . Let us consider, for each
point w of the sphere, a r-frame consisting of the vector w and the fixed (r − 1)-frame,
one obtains an element of Vr (Cn ). The induced map from the oriented sphere S2p−1 to
Vr (Cn ) defines a generator of π2p−1 (Vr (Cn )).
One obtain the following result:
Proposition 3.19 – Real case. Let us consider an r-frame v (r) defined on the boundary
of the k-cell d.
32
• If k < n − r + 1, one has [ξ(v (r) , d)] = 0 and one can extend the r-frame defined on ∂d
inside d without singularity.
• If r = 1 and k = n, one can extend the vector field v (1) = v defined on ∂d inside d with
ˆ
an isolated singularity in the barycenter dˆ of the cell, with index [ξ(v, d)] = I(v, d).
That is the index we defined in Definition 2.1.
• If r > 1 and k = n − r + 1, then one can extend the r-frame v (r) defined on ∂d inside
d with an isolated singularity at the barycenter dˆ of the cell. In that case, [ξ(v (r) , d)]
is an integer if k is odd and an integer mod 2 if k is even. Reducing modulo 2,
ˆ that measures the obstruction to the extension of v (r)
one obtains an index I(v (r) , d)
inside the k-cell d.
The dimension p = n − r + 1 is called the obstruction dimension for the construction of
a r-frame tangent to M .
Proposition 3.20 – Complex case. Let us consider a complex r-frame v (r) defined on
the boundary ∂d of the k-cell d.
• If k < 2(n − r + 1), one has [ξ(v (r) , d)] = 0 and one can extend the r-frame v (r) inside
d without singularity.
• If k = 2(n−r +1), then one can extend the r-frame v (r) inside d with an isolated singularity at the barycenter dˆ of the cell. In that case, one obtain an index [ξ(v (r) , d)] ∈ Z
ˆ The index measures the obstruction to the extension of
that we define as I(v (r) , d).
(r)
v , defined on the boundary ∂d, inside d.
The dimension 2p = 2(n − r + 1) is called the obstruction dimension for the construction
of a complex r-frame tangent to M .
3.4
Applications: Stiefel-Whitney and Chern classes
Let us apply the previous construction to the cases of r-fields tangent to a manifold, in
the real and the complex case.
3.4.1
Stiefel-Whitney classes
The Stiefel-Whitney classes have been defined by obstruction theory (see [Sti],[Wh1]). In
fact, Whitney used the same strategy than Stiefel, applying it to arbitrary sphere bundles.
We use the Steenrod construction ([Ste], part III).
The pth Stiefel-Whitney class of M , denoted by wp (M ), is defined as the primary
obstruction to constructing an r-frame over M , that is a section of Vr (T M ) or a set of r
linearly independent vector fields tangent to M , with p = n − r + 1. More precisely, one
performs the following construction:
Using the result in (3.17) one can construct an r-frame by choosing any r-frame v (r)
on the 0-skeleton of the cell decomposition (D), then extending it without zeroes till the
obstruction dimension p = n − r + 1. That means that v (r) has no singularity on the
33
(p − 1)-skeleton and isolated singularities on the p-skeleton of (D). Given the r-frame
v (r) on the boundary of each p-cell d, one can extend v (r) on d with a singularity at the
barycenter dˆ of index
(
ˆ = [(v (r) )p−1 |∂dp ] ∈ πp−1 (Vr (Rn )) = Z for p = n − r + 1 odd or p = n if r = 1
I(v (r) , d)
Z2 for p = n − r + 1 even and r > 1,
using the notation in 3.6. In any case, there is a non trivial homomorphism from πp−1 (F )
ˆ ∈ Z2 .
to Z2 . hence we can reduce the coefficients
modulo 2 obtaining I(v (r) , d)
P
(r) ˆ ∗
p
One can define the p-cochain
I(v , d) d in C (D, Z2 ), its value on each p-cell d
(r) ˆ
is I(v , d). According to Theorem 3.8, the cochain is in fact a cocycle and defines an
element wp (M ) in H p (M ; Z2 ). The Definition 3.13 provides the following:
Definition 3.21 The p-th Stiefel-Whitney class of M , denoted by wp (M ) ∈ H p (M ; Z2 )
is the class of the primary obstruction cocycle corresponding to constructing an r-frame
tangent to M .
By the general obstruction theory, the obtained classes do not depend on the choices
we make in the construction.
In the particular case r = 1, one can use integer coefficients. The evaluation of
m
w (M ) ∈ H m (M ; Z) on the fundamental class [M ] of M is the Euler-Poincaré characteristic of M .
Let us suppose that the cell decomposition (D) is obtained by duality of a triangulation
(K) of M . Each p-cell d = d(σ) in (D) is dual of an (r − 1)-simplex σ in (K). By Poincaré
duality (cap-product by the fundamental class),
H m−r+1 (M ; Z2 ) −→ Hr−1 (M ; Z2 )
the image of d∗ is σ and image of wp (M ) is the so-called (r − 1)-homology Siefel-Whitney
class, denoted by wr−1 (M ). A cycle representing wr−1 (M ) is given (mod 2) by
X
ˆ
I(v (r) , d(σ))σ.
dim σ=r−1
In fact, Stiefel-Whitney classes can be defined in a combinatorial way [HT].
3.4.2
Chern classes
The definition of Chern classes by obstruction theory in the complex case is similar to the
real case, even simpler.
Let M denote an analytic complex manifold and T M the complex tangent bundle
to M . The pth Chern class of M , denoted by cp (M ), will be defined as the primary
obstruction to constructing a complex r-frame over M , that is a section of Vr (T M ) or a
set of r linearly independent vector fields tangent to M , with p = n − r + 1.
Using the result in (3.18) one can construct an r-frame by choosing any r-frame v (r)
on the 0-skeleton of the cell decomposition (D), then extending it without zeroes till the
obstruction dimension
2p = 2(n − r + 1).
(3.22)
34
That means that v (r) has no singularity on the (2p − 1)-skeleton and isolated singularities
on the 2p-skeleton of (D). Given the r-frame v (r) on the boundary of each 2p-cell d, one
can extend v (r) on d with a singularity at the barycenter dˆ of index
ˆ = [(v (r) )2p−1 |∂d2p ] ∈ π2p−1 (Vr (Cn )) = Z
I(v (r) , d)
using the notation in 3.6.
The generators of π2p−1 (Vr (Cn )) being consistent (see [Ste]), one can define the 2pP
ˆ d∗ in C 2p (D, Z), its value on each 2p-cell d is I(v (r) , d).
ˆ this defines a
cochain
I(v (r) , d)
cochain
γ ∈ C 2q (M ; π2q−1 (Wr,m )) ,
by γ(d) = I(v (r) , d), for each 2q-cell d, and then extend it by linearity.
One can define the 2p-cochain
X
ˆ d∗ ∈ C 2p (K, Z)
I(v (r) , d)
(3.22)
ˆ According to Theorem 3.8, the cochain is in
whose value on each 2p-cell d is I(v (r) , d).
fact a cocycle and defines an element cp (M ) in H 2p (M ; Z). The Definition 3.13 provides
the following:
Definition 3.23 The p-th Chern class of M , denoted by cp (M ) ∈ H 2p (M ; Z) is the
class of the primary obstruction cocycle corresponding to constructing a complex r-frame
tangent to M .
By the general obstruction theory, the obtained classes do not depend on the choices
we make in the construction.
In the particular case r = 1, the evaluation of cm (M ) on the fundamental class [M ] of
M yields the Euler-Poincaré characteristic of M .
Note that cm (M ) coincides with the Euler class of the underlying real tangent bundle
TR M , so these classes are natural generalization of the Euler class.
Let us suppose that the cell decomposition (D) is obtained by duality of a triangulation
(K) of M . Each 2p-cell d = d(σ) in (D) is dual of an 2(r − 1)-simplex σ in (K). By
Poincaré duality (cap-product by the fundamental class),
H 2(m−r+1) (M ; Z) −→ H2(r−1) (M ; Z)
the image of d∗ is σ and image of cp (M ) is the so-called 2(r − 1)-homology Chern class,
denoted by cr−1 (M ). A cycle representing cr−1 (M ) is given by
X
ˆ
I(v (r) , d(σ))σ.
(3.23)
dim σ=2(r−1)
As we will see in the next sections, there is no cohomology Chern class in the case of
singular varieties, but expression (3.23) will generalise for suitable frames we will define.
35
3.5
Axiomatic definition
Definition 3.24 [Theorem] To each complex vector bundle ξ of rank n over a space M ,
one can associate a class c(ξ) = 1 + c1 (ξ) + · · · + cn (ξ), where ci (ξ) ∈ H 2i (M ; Z) and
ci (ξ) = 0 if i > n, satisfying the following properties:
1. (Naturality) For each f : Y → M , then f ∗ (c(ξ)) = c(f ∗ (ξ)).
2. (Whitney sum) If ξ and η are two bundles over M , then
c(ξ ⊕ η) = c(ξ) ∪ c(η).
3. the class c1 (γ11 ) ∈ H 2 (CP1 ; Z) of the canonical line bundle over CP1 (see 3.4) is non
zero.
Modulo naturality, the last axiom is equivalent to the following ones:
40 . Let γ1n be the canonical bundle over CPn , then c1 (γ1n ) = hn is a generator of
H 2 (CPn ; Z) (see 3.4).
400 . Let γ1 be the canonical line bundle over CP∞ , then c1 (γ1 ) is a generator of the
polynomial ring H ∗ (CP∞ ; Z).
Proposition 3.25 If ξ is a trivial bundle, then ci (ξ) = 0 for i ≥ 1.
36
4
Hirzebruch theory
In this section, one explicits the Hirzebruch theory that provides a way to unify, in the
case of manifolds, the three theories of characteristic classes: the Chern class, the Todd
class and the Thom-Hirzebruch class.
4.1
The arithmetic genus
Let gi be the number of C-linearly independent holomorphic differential i-forms on the
n-dimensional complex algebraic manifold X. The number gi equals the Hodge number
hi,0 (M ).
• g0 is the number of linearly independent holomorphic functions, i.e. the number of
connected components of X,
• gn is called geometric genus of X,
• g1 is called irregularity of X,
Definition 4.1 [Arithmetic Genus] The arithmetic genus of X, denoted by χa (X) is
defined as :
n
X
χa (X) :=
(−1)i gi
i=0
Example 4.2 Let X be a complex algebraic curve, i.e. a compact Riemann surface. X
is homeomorphic to a sphere with g handles. Then g0 = 1 and g1 = gn = g.
The arithmetic genus of X is:
χa (X) = 1 − g
4.2
The Todd genus
The Todd genus T (X) has been defined (by Todd) in terms of Eger-Todd fundamental
classes (polar varieties), using Severi results. The Eger-Todd classes are homological
Chern classes of X.
Todd “proved” that
T (X) = χa (X).
In fact, the Todd proof uses a Severi Lemma which has never been completely proved.
The Todd result has been proved by Hirzebruch, using other methods.
37
4.3
The signature
Definition 4.3 [Thom-Hirzebruch] Let M be a (real) compact oriented 4k-dimensional
manifold. Then the map
H 2k (M ; R) × H 2k (M ; R) −→ R,
(x, y) 7→ hx ∪ y, [M ]i ∈ R
defines a bilinear form on the vector space H 2k (M ; R).
The index (or signature) of M , denoted by sign(M ), is defined as the index of this
form, i.e. the number of positive eigenvalues minus the number of negative eigenvalues.
4.4
Hirzebruch Theory
The Hirzebruch theory uses multiplicative series and Chern root. let us define (or recall)
these two ingredients.
4.4.1
Hirzebruch Series
For y ∈ R, let us define the Hirzebruch multiplicative series (for a review on multiplicative
series, see Hirzebruch [Hirz]):
Qy (α) :=
α(1 + y)
− αy ∈ Q[y][[α]]
1 − e−α(1+y)
One has the particular cases:
•
Q−1 (α) = 1 + α
•
Q0 (α) =
•
Q1 (α) =
4.5
y = −1
α
y=0
α
tanh α
y=1
1 − e−α
Characteristic Classes of Manifolds
Let X be a complex manifold with dimension dimC X = n, let us denote by
∗
c (T X) =
n
X
cj (T X),
cj (T X) ∈ H 2j (X; Z)
j=0
the total Chern class of the (complex) tangent bundle T X.
Definition 4.4 The Chern roots αi of the complex manifold X are elements in H 2 (X; Z)
defined by:
n
n
X
Y
cj (T X) tj =
(1 + αi t).
j=0
i=1
38
g
For each y ∈ R, one defines the Todd-Hirzebruch class: td
(y) (T X) :=
n
Y
Qy (αi )
i=1



c∗ (T X)


















td∗ (T X)
g
td(y) (T X) =















L∗ (T X)






n
Y
=
(1 + αi )
y = −1
i=1
Chern class,
=
n
Y
i
( 1−eα−α
)
i
y=0
i=1
Todd class,
n
Y
αi
)
=
( tanhα
i
y=1
i=1
Thom-Hirzebruch L-class.
Remark 4.5 The previous equalities can be considered as definition of the Todd class
and the Thom-Hirzebruch L-class for the reader who is not already aware of these notions.
4.6
The χy -characteristic
Let X be a complex projective manifold.
Definition 4.6 For each y ∈ R, one defines the χy -characteristic of X by
χy (X) :=
<∞
<∞
X
X
p=0
!
(−1)i dimC H i (X, ∧p T ∗ X
· yp.
i=0
One has the particular cases:
• y = −1
χ−1 (X) = χ(X), Euler-Poincaré characteristic of X (by Hodge theory),
• y=0
χ0 (X) = χa (X), arithmetic genus of X (by definition),
• y=1
χ1 (X) = sign(X), signature of X (by Hodge theory).
One has the following table of invariants:
y
χy (X)
g
td
(y) (T X)
-1
χ(X)
c∗ (T X)
0
χa (X)
td∗ (T X)
1
sign(X)
L∗ (T X)
39
4.7
Hirzebruch Riemann-Roch Theorem
Theorem 4.7 (Hirzebruch Riemann-Roch Theorem) One has:
Z
g
χy (X) =
td
∈ Q[y].
(y) (T X) ∩ [X]
X
In the three particular cases, one obtains:
R
• χ(X) = X c∗ (T X) ∩ [X]
y = −1
Euler - Poincaré characteristic of X
Poincaré-Hopf Theorem,
• χa (X) =
R
X
td∗ (T X) ∩ [X]
y=0
arithmetic genus of X
Hirzebruch-Riemann-Roch Theorem,
• sign(X) =
R
X
L∗ (T X) ∩ [X]
y=1
signature of X
Hirzebruch signature Theorem.
40
5
Singular varieties
A singular variety is a variety which contains singular points, that is points for which the
property in Definition 1.8 is not satisfied. Examples of singular varieties are the following:
The pinched torus: the pinched point a does not admit any neighbourhood satisfying the
property 1.8. In that case, the link of an “elementary neighbourhood” of a is the union of
two not connected circles. Another example is provided by the suspension of the torus.
The two points a and b of the suspension of the torus are singular points, in that case,
the link of a (or b) is a torus, it is not a sphere.
In order to extend the notion of characteristic classes to singular varieties, it is necessary to know the local structure of the singular variety. That is given by the structure of
stratified space and by suitable definition of triangulation on the variety.
5.1
Stratifications
Definition 5.1 Let X be a topological space, we denote by X a filtration of X by closed
subsets
∅ = X−1 ⊂ X0 ⊂ X1 ⊂ · · · ⊂ Xn−2 ⊂ Xn−1 ⊂ X = Xn
(5.2)
A topological stratification of X is the data of a filtration X of X such that each difference
Vk = Xk − Xk−1 is either empty or a topological manifold of pure dimension k. The
connected components of the Vk are called the strata.
The stratifications that we will consider will be locally finite partitions of X into locally
closed submanifolds, the strata, satisfying the frontier condition:
Vk ∩ V j 6= ∅ ⇒ Vk ⊂ V j
Let X be a closed subset of a differentiable manifold M . A differentiable stratification
of X is a topological stratification X of X such that each stratum in Vk is a differentiable
submanifold of M .
In order to work with, the considered stratification should satisfy conditions which
precise the way the strata are glued together. On one hand, there are many ways to
define these conditions, according to the specific problem. On the other hand, given
conditions on the stratification, one has to know what kind of singular variety admits a
stratification satisfying these conditions. In the following, one considers the stratifications
which will be useful for the construction of characteristic classes. One refer to [B3, Tr]
for more information on the different types of stratifications.
41
5.1.1
Whitney stratifications
As we have seen, on a singular variety, there is no longer tangent space in the singular
points. One way to find a substitute for the tangent bundle is to stratify the singular
variety in submanifolds. One can proceed the following construction: If X is a singular
analytic variety, equipped with a stratification and embedded in a smooth analytic manifold M , one can consider the union of tangent bundles to the strata, that is a subspace E
of the tangent bundle to M . The space E is not a bundle but it generalizes the notion of
tangent bundle in the following sense: A section of E over X is a section v of T M |X such
that at each point x ∈ X the vector v(x) belongs to the tangent space of the stratum
containing x. Such a section is called a stratified vector field over X. To consider E as
the substitute for the tangent bundle of X and to use obstruction theory is the M.-H.
Schwartz point of view (1965, [Sc2]) in the case of analytic varieties.
When one consider stratification of singular varieties, it is natural to ask for conditions
with which the strata glue together. The so-called Whitney conditions are those which
allow one to proceed to the construction of radial extension of vector fields. According to
a result of Whitney, every analytic variety can be equipped with a Whitney stratification.
Definition 5.3 One says that the Whitney conditions are satisfied for a stratification if,
for any pair of strata (Vi , Vj ) such that Vi is in the closure of Vj , one has:
a) if (xn ) is a sequence of points in Vj with limit y ∈ Vi and if the sequence of tangent
spaces Txn (Vj ) admits a limit T (in the suitable grassmanian space) when n goes to +∞,
then Ty (Vi ) is included in T .
b) if (xn ) is a sequence of points in Vj with limit y ∈ Vi and if (yn ) is a sequence of points
in Vi with limit y, such that the sequence of tangent spaces Txn (Vj ) admits a limit T for
n going to +∞ and such that the sequence of directions xn yn admits a limit λ when n
goes to +∞, then λ lies in T .
(a) Condition
(b) Condition
Figure 6: Whitney conditions
Example The conditions (a) and (b) are not satisfied for the stratification of the cone X
consisting of a generatix D = Vi and Vj = X \ D. This is clear taking for (xn ) a sequence
of points going to the vertex y of the cone, along a generatrix (different from D), and for
42
yn a sequence of points such that the segment xn yn has always the same direction. Adding
the vertex of the cone as a supplementary 0-dimensional stratum, the new stratification
satisfies the Whitney conditions.
Figure 7: Whitney conditions on the cone
Example Let V be the variety whose equation in C3 is y 2 + x3 − t2 x2 = 0, stratified by
the horizontal axis W = Vi and Vj = V − Vi , then the (a)-condition is satisfied but not
(b). Adding the vertex of Cn as a new stratum, the Whitney conditions are verified.
Figure 8: Whitney conditions
Let us consider a neighborhood of a point y ∈ Vi , and a sufficiently samlll Let Dε
a disk in M transverse to Vi at y and let us consider at x ∈ Vj ∩ D the vector ṽ(x)
parallel to v(y), For epsilon sufficiently small, angle(ṽ(x), Tx (Vj )) is small (less than ε.
The projection of ṽ(x) on Tx (Vj )) does not vanish.
5.2
Poincaré homomorphism
Let us recall (see [B1]) that the Poincaré homomorphism can be described in the following
way. Let us suppose that the oriented singular n-dimensional variety X is a subvariety
of a P.L. oriented m-dimensional manifold M . Any stratification of X defines a stratification of M adding M − X as regular stratum. Let us denote by (K) a locally finite
triangulation of M compatible with the stratification and by (K 0 ) a barycentric subdivision of (K). The chain (or cochain) complexes relatively to (K) or (K 0 ) will be denoted
(K 0 )
∗
by C∗ (X), C(K)
(X) for example.
43
Providing to all n-dimensional simplexes of (K 0 ) the orientation of Xreg , the sum of
(K 0 )
these simplexes is a cycle in Cn (X). Its class in Hn (X) is the fundamental class of X,
denoted by [X].
For every (n − i)-simplex σ in (K), the dual cell of σ in M , denoted by d(σ) has
dimension m − (n − i). It is transverse to X, i.e. to every stratum Xn−α − Xn−α−1 of X.
The intersection d(σ) ∩ X is an oriented i-dimensional (K 0 )-chain in X.
Let us define the Poincaré homomorphism
H n−i (X) → Hi (X)
given by the chain map
(K 0 )
n−i
C(K)
(X) → Ci
(X)
which maps the elementary (n − i)-cochain d∗ = d∗ (σ), dual of the simplex σ in K, to the
i-chain ξ = d(σ) ∩ X of K 0 . In other words, ξ is the cap-product of σ ∗ by [X] (see [B1]).
5.2.1
Alexander isomorphism
The Alexander isomorphism (see [B1]) H m−i (M, M − X) → Hi (X) is induced by the
isomorphism:
(K)
m−i
C(D)
(M, M − T̊ ) → Ci (X)
which associates to a D-cochain (dm−i )∗ such that dm−i ∩ X 6= ∅ the K-chain σi such that
dm−i = d(σi ).
5.3
Poincaré-Hopf Theorem: The singular case
If X is a singular variety, the Poincaré-Hopf Theorem fails to be true. The main reason
is that there is no longer tangent space at each singularity. The definition of the index of
a vector field in one of its singular points takes sense on a smooth m-manifold only. In
particular the singular point must have a neighbourhood isomorphic to the ball Bm and
whose boundary is isomorphic to the sphere Sm−1 . Let us consider the example of the
pinched torus X in R3 . The pinched point a is a singular point of X, in fact it constitutes
the singular part of the pinched torus. The only ‘natural’ way to define an index of a
vector field at the point a is to consider a vector field v defined in a ball B 3 (a) centered in
a, in R3 , with an isolated singularity at a, such that if x ∈ X \ {a}, then v(x) is tangent to
the smooth manifold X \ {a} and such that v does not have other singularities in B 3 (a).
Let us consider two examples of such vector fields:
a) The vector field tangent to the parallels of the torus T (see Figure 9a) determines,
on the pinched torus X, a vector field v pointing inward the ball B 3 (a) along one of the
two unlinked circles, which are the intersection of ∂B 3 (a) and X, and pointing outward
the ball along the other circle. On the one hand, this vector field, defined on ∂B 3 (a)∩X, is
the restriction of a vector field w defined on ∂B 3 (a) with index 0 at a. On the other hand,
there is no more singularity of v on X \ {a}. In this case, the Poincaré-Hopf Theorem is
not satisfied: one has
χ(X) = 1 6= 0 = I(w, a).
44
b) Let us consider a vector field v pointing outward the ball B 3 (a) along ∂B 3 (a) and
tangent to X along the restriction ∂B 3 (a) ∩ X (see Figure 9b) . This vector field has
index +1 at a. It is orthogonal to the two meridians ∂B 3 (a) ∩ X and it can be extended
on the pinched torus as a continuous vector field without other singularity. In fact, one
can define an extension v such that, on each meridian, the angle of v(x) with the tangent
space to the meridian is constant and this angle goes down continuously as the distance
to a grows till being 0 for the meridian opposed to a. In this case, the Poincaré-Hopf
Theorem is valid:
χ(X) = 1 = I(v, a).
(a) The vector field v tangent
to the parallels
(b) The radial vector field
Figure 9: Vector fields on the pinched torus
The radial vector field in case (b) is the first example of M.-H. Schwartz’s radial vector
field, of which we will make a systematic study.
If X is a singular variety, the Poincaré-Hopf Theorem fails to be true, the main reason
is that there is no more tangent space in singular points. The definition of the index of a
vector field at one of its singular points takes sense on a smooth manifold only, the reason
being that the link of a point is a sphere.
In order to obtain a Poincaré-Hopf Theorem, one can think to consider a stratification
of the singular variety (see section 5.1), i.e. a decomposition of the singular variety into
smooth manifolds and consider continuous vector fields which are stratified.
Definition 5.4 A stratified vector field v on X is a (continuous) section of the tangent
bundle of M , T (M ), such that, for every x ∈ X, then one has v(x) ∈ T (Vi(x) ) where Vi(x)
is the stratum containing x.
45
Proposition 5.5 [Sc4] If the stratification satisfies Whitney conditions (a) and (b), then
there exists on X a stratified vector field v with isolated singularities ak . The index
I(v|Vi(ak ) , ak ), defined in the stratum Vi(x) containing x, is well defined.
One could define the index of a stratified vector field v at a singular point a situated
in the stratum Vi as the index of the restriction I(v|Vi , a). The natural generalization of
the Poincaré-Hopf Theorem to singular varieties would be the following formula:
X
χ(X) =
I(v|Vi(ak ) , ak ).
(5.6)
ak ∈Sing(v)
In general, the formula(5.6) is not true. Let us provide the (counter)-example given
in [Sc4, 6.2.1]:
Example 5.7 In a first step, in R2 with coordinates (x, y), one considers the (closed)
balls centered at the origin, B with radius 1 and D with radius 2. One has χ(D) = +1.
Figure 10: Counter exemple for Poincaré-Hopf
Inside the ball B, one considers the continuous vector field v1 (x, y) = (|x|, y). One has
v1 (0) = 0, the point 0 is a singularity of v1 with index I(v1 , 0) = 0.
On the boundary ∂D, one considers the vector field v2 (x, y) = (x, y) pointing radially
outwards. One can extend v2 inside D as a continuous vector field v which is v2 along
∂D, v1 inside B and which is tangent to the y-axis Y along Y . The vector field defined
by
(
p
2
2
2|x| − x + (x − |x|) x + y , y
on D \ B
v(x, y) =
v1 (x, y) = (|x|, y)
inside B
46
satisfies the conditions.
The vector field v has an isolated singular point of index 0 at 0 and another isolated
singular point at a = (−3/2, 0) ∈ D\B. By Poincaré-Hopf Theorem with boundary(2.4.3),
one has
χ(D) = +1 = I(v, 0) + I(v, a),
that implies I(v, a) = +1.
Let us remark that while I(v, 0) = 0, one has I(v|Y , 0) = +1.
Now fold the picture along the y-axis, as a (differentiable) singular surface x2 − z 3 = 0
in R3 . In that case, D becomes a singular variety ∆, with boundary and stratified by Y
and ∆ \ Y . The vector field v in D defines a stratified vector field, still denoted by v in ∆.
It has two isolated singular points: 0 and a. One has I(v|Y , 0) = +1 and I(v, a) = +1.
The formula (5.6) would be written:
χ(∆) = +1 6= I(v, a) + I(v|Y , 0) = 1 + 1 = 2.
Let us remark that the vector field v is not “radial” at the singular point 0, it is not
pointing outwards the unit ball centered at 0.
The main result of M.H. Schwartz is to provide, for singular varieties, an explicit
construction of certain vector fields, called radial vector fields for which the PoincaréHopf theorem is still valid.
Moreover, in the same way that the radial vector fields allow to recover the PoincaréHopf Theorem, the construction of characteristic classes for singular varieties will consist
in a construction of vector frames adapted to the singular situation and generalising the
notion of radial vector fields.
5.3.1
Radial extension process
One gives a description of the local radial extension process. This will be used for the
global process in the next section.
The local radial extension, defined by M.-H. Schwartz, is similar to the one defined by
Milnor in order to prove Theorem 2.13.
Let X be a singular variety embedded in an m-dimensional manifold M . Let Vα ⊂ X
be a stratum, B(a) ⊂ Vα a neighbourhood of the point a in Vα and v a vector field defined
on B(a) with an isolated singularity at a.
One can construct two vector fields:
1. Parallel Extension. We will denote by N (a) a tubular neighbourhood homeomorphic
to B(a) × Dk , where Dk is a disk transverse to Vα and k = m − dim Vα . Let us consider
the parallel extension v̂ of v in the tube N (a). Let Vβ be a stratum such that a ∈ Vβ . At
a point x ∈ N (a) the parallel extension v̂(x) is not necessarily tangent to Vβ . However,
the Whitney condition (a) guarantees that if N (a) is sufficiently small, then the angle
between Ta (Vα ) and Tx (Vβ ) is small. That implies that the orthogonal projection of v̂(x)
onto Tx (Vβ ) does not vanish. Of course, considering for each stratum the projection of
the parallel extension on the tangent space to the stratum at the given point does not
provide a continuous vector field. In order to obtain a continuous vector field, one has
47
to consider a slight modification of the construction in the neighbourhood of the strata,
which is easy to understand, but complicated to describe in details. A good extension
will be v̂(x) away from Vβ and continuously going to the projection of v̂(x) onto Tx (Vβ )
when approaching Vβ , using a suitable partition of unity. The construction of such an
extension is correctly and entirely described in M.-H. Schwartz’s book [Sc4]. In fact, one
has to work simultaneously for all strata Vβ such that a ∈ Vβ , that complicates a detailed
construction.
In conclusion, the Whitney condition (a) implies that one can proceed to the construction of a stratified vector field, still denoted by v̂(x), which is a “parallel extension” of the
given vector field on Vα , in a suitably small tubular neighbourhood around B(a) in M .
One observes that the singular locus of v̂ corresponds to an (m − dim Vα )-dimensional
disk which is transversal to B(a) in M .
Figure 11: The local radial extension of a vector field
2. Transversal vector field. Let us consider the transversal vector field τ (x), as in the
proof of Theorem 2.13. This vector field is essentially the gradient of the function “square
of the distance to Vα ”, for an appropriate Riemannian metric. The vector field τ (x) is not
necessarily tangent to the strata Vβ such that a ∈ Vβ . However, the Whitney condition
(b) guarantees that in a sufficiently small “tube” around B(a), the angle between τ (x)
and Tx (Vβ ) is small. That means that the orthogonal projection of τ (x) onto Tx (Vβ ) does
not vanish. In the same way as for the parallel extension, for each stratum one could
consider the projection of τ (x) onto the tangent space to the stratum at the given point.
However, this does not provide a continuous vector field. In order to obtain a continuous
vector field, one has to consider a similar modification of the construction. The correct
vector field is τ (x) away from Vβ and continuously going to the projection of τ (x) onto
Tx (Vβ ) when approaching Vβ . That construction is also completely described in M.-H.
Schwartz’s book [Sc4], and one has to work simultaneously for all strata Vβ such that
a ∈ Vβ .
48
In the boundary of the tube N (a) ∼
= B(a) × Dk the part B(a) × ∂Dk = B(a) × Sk−1
shall be called the horizontal part.
In conclusion, one obtains a stratified “transversal” vector field, still denoted by τ (x),
which is zero along Vα and growing with the distance to Vα and which is pointing outward
the horizontal part of the boundary of the tube N (a) provided that the tube is sufficiently
small.
Definition 5.8 The radial extension of the vector field v defined on B(a) ⊂ Vα is the
vector field ṽ defined on the tube N (a) as the sum
ṽ(x) = v̂(x) + τ (x).
Proposition 5.9 The radial extension ṽ of the vector field v is transversal to the boundaries of the tube N (a) around B(a), pointing outward the horizontal part of ∂N (a). Its
unique singularity inside N (a) is a, i.e. the same singularity as the initial vector field v.
Moreover, the index of ṽ in a, computed in the tube N (a), is the same as the index of v
at a, computed in the manifold Vα , namely we have
I(ṽ, a; M ) = I(v, a; Vα ).
This property, i.e., the above equality, is the main property of the radial extension,
that is precisely the property which allows one to prove the Poincaré-Hopf Theorem for
singular varieties.
5.3.2
Poincaré-Hopf Theorem for singular varieties.
In this section one proceeds to the construction of a “global” radial extension of a vector
field, which M.-H. Schwartz called radial vector field and one shows the following:
Theorem 5.10 Let X ⊂ M be a (compact) singular variety embedded in a manifold M .
One can construct on X a (stratified) radial vector field, in the sense of M.-H. Schwartz.
That is a vector field v defined in a tubular neighbourhood Nε (X) of X in M , pointing
outward Nε (X) along the boundary. It has finitely many isolated singularities ai in Nε (X),
all situated in X, and one has
I(v, ai ; M ) = I(v|Vα(i) , ai ; Vα(i) ),
where Vα(i) is the stratum of X containing ai .
Proof: The “global” construction of the radial vector field goes as follows:
One consider a Whitney stratification on X as before and one adds the stratum M \ X
in order to obtain a Whitney stratification of M .
The aim of the process is to construct, by increasing induction on the dimension of
the strata of X, a stratified vector field v in a neighbourhood of X in M , with finitely
many isolated singularities ai in the strata Vα , such that if ai ∈ Vα , the index of v at ai is
the same, computed in Vα or in M .
By the induction process on the dimension of the strata, we will show the following:
49
(P) For each stratum Vα , there is a neighbourhood Nεα around V α and a stratified
vector field v defined on Nεα , pointing outward Nεα along the boundary, with isolated
singularities ai in V α and such that if Vβ is the stratum containing ai , then the index
of ai computed in Vβ is the same as the index of ai computed in Nεα , i.e. in M .
Namely, we will denote
I(v, ai ) = I(v, ai ; Vβ ) = I(v, ai ; M ).
The neighbourhood Nεα is the set of points in M with distance less than ε from points
in V α . That is not a fibre bundle over V α , but the following construction shows that
for each stratum Vβ ⊂ V α , there is a neighbourhood Aβ of V β \ Vβ in Vβ such that
the restriction of Nεα to Aβ is a fibre bundle with fibre a disk whose dimension is the
codimension of Vβ in M .
Let us show that induction property (P) is true for the lowest dimensional stratum.
If the lowest dimensional strata in X are 0-dimensional ones, i.e. V0 is a set of finitely
many points ak , then one considers a radial vector field v in a ball Bε (ak ) centered at
each of these points. According to the Bertini-Sard Lemma, the boundary ∂Bε (ak ) is
transverse to the strata Vγ containing ak in their closure (one takes for ε the smallest of ε
for all ak ). For each point x ∈ Vγ ∩ ∂Bε (ak ), the radial vector field v(x) is not orthogonal
to Tx (Vγ ). One can deform the radial vector field to a stratified vector field v pointing
outward Bε (ak ) along ∂Bε (ak ). In this case, the index I(v, ak ) is +1 and obviously one
has
X
χ(V0 ) =
I(v, ak ).
k
In this case, Nε0 is the union of Bε (ak ).
If the lowest dimensional stratum is a stratum Vα of dimension s > 0, then one
constructs, by classical obstruction theory, a vector field v on Vα with finitely many
isolated singularities ai . We notice that Vα is a manifold without boundary and it has to
be compact if X is compact. According to the classical Poincaré-Hopf Theorem 2.14, one
has
X
χ(Vα ) =
I(v, ai ).
i
The extension process in a neighbourhood Nεα of Vα , described in the proof of Theorem
2.13, can be slightly deformed according to the local case (cf Proposition 5.9) in order to
obtain a stratified vector field defined on Nεα , pointing outward Nεα along its boundary
and such that the index of v at each singularity ai is the same, whether it is computed in
Vα or in Nεα , i.e. in M .
Let us now suppose that induction property (P) holds for all strata up to Vα (i.e. for
all strata whose dimension is less than or equal to dim Vα ) and let us call Vγ the following
one. One has to show that (P) holds for Vγ .
The vector field v is defined on Uγ = Vγ ∩ Nεα and is pointing inward Vγ along ∂Uγ =
U γ \ Uγ . By classical obstruction theory, one can extend v inside Vγ , as a vector field still
denoted by v, with finitely many isolated singularities aj in Vγ \ Uγ .
For t ∈]0, 1], let us denote by Ntεα the (open) neighbourhood of V α , which is the set of
α
can
points whose distance to V α is less than tε. The vector field v defined on Vγ \ Nε/2
50
Figure 12: The radial vector field
be extended, using the local extension process, as a vector field v 0 defined in a tube Nη0
α
without other singularities than the points aj and such that
around Vγ \ Nε/2
I(v, aj ; Vγ ) = I(v, aj ; Nη0 ).
α
and the fiber a disk whose
Let us notice that Nη0 is a disk bundle with the basis Vγ \ Nε/2
dimension is the codimension of Vγ in M .
On the intersection Nεα ∩ Nη0 , one has two vector fields: v defined on Nεα and v 0 defined
on Nη0 . They coincide on Vγ . The vector field defined by w = (2 − 2t)v + (2t − 1)v 0 on
α
and with v 0 on ∂Nη0 .
each Ntεα , for t ∈ [1/2, 1], coincides with v on ∂Nε/2
α
, v 0 on Nη0 \ Nεα and w on Nεα ∩ Nη0 satisfies the
The vector field defined as v on Nε/2
α
∪ Nη0 .
property (P) for Vγ with the neighbourhood Nεγ defined as Nε/2
At the last step, one denotes by Nε (X) the neighbourhood of X constructed by the induction process. By construction, the stratified radial vector field v satisfies the statement
of the theorem.
Theorem 5.11 (Poincaré-Hopf Theorem for singular varieties) Let X ⊂ M be a (compact) singular variety embedded in a manifold M , and v be a stratified radial vector field
defined in the neighbourhood Nε (X) of X (Theorem 5.10). Then one has
X
χ(X) =
I(v, ai ).
i
Proof:
Using the stratified radial vector field constructed in the proof of Theorem
5.10 and Theorem 2.18, one has
X
χ(Nε (X)) =
I(v, ai ).
i
51
Note that X is a deformation retract of its neighbourhood Nε (X), that ends the proof.
Here we remark that at each stage of the proof of Theorem 5.10 one has
X X
I(v, ai ),
χ(V α ) =
Vβ ⊂V α ai ∈Vβ
the first summand being for all strata Vβ in V α , including Vα .
In fact, our proof shows the more precise result:
Theorem 5.12 ([Sc4, Théorème 6.2.2]) Let X be an analytic subset of the analytic manifold M and {V /α} a Whitney stratification of the pair (M, X). Let us denote by D a
compact domain with a smooth boundary transverse to the strata. Let v (resp. v − ) be
a radial vector field pointing outwards (resp. inwards ∂D). There is a finite number of
zeroes ai of v (resp. v − ) in D and we have :
X
X
χ(X ∩ D) =
I(v, ai ) =
I(v|Vα(ai ) , ai )
ai ∈X∩D
χ(X ∩ D) − χ(X ∩ ∂D) =
X
ai ∈X∩D
χ(Vα ∩ int(D)) =
Vα ⊂X
=
X
ai ∈X∩D
−
I(v |Vα(ai ) , ai ),
ai ∈X∩D
where, if dim Vα(a) = 0, then by convention I(v|Vα(a) , a) = +1.
52
X
I(v − , ai )
6
Schwartz and MacPherson classes
In the following, M will be a complex analytic manifold equipped with an analytic stratification {Vi }: for every stratum Vi , the closure V̄i and the boundary V̇i = V̄i \ Vi are
analytic sets, union of strata. We denote by X ⊂ M a complex analytic compact subset
stratified by {Vi }.
The first definition of Chern class for singular varieties was given by M.H. Schwartz in
the preprint [Sc1] (Lille University), then in 1965 in two “Notes aux CRAS” [Sc2]. Here
we provide a sketch of the M.H. Schwartz construction.
6.1
Radial frames
Let X ⊂ M be a singular n-dimensional complex variety embedded in a complex mdimensional manifold. Let us consider a Whitney stratification {Vi } of M such that
X is a union of strata and denote by (K) a triangulation of M compatible with the
stratification, i.e. each open simplex is contained in a stratum.
The first observation by M.-H. Schwartz concerns the triangulations:
One knows (3.22) that the primary obstruction dimension to the construction of an
r-frame tangent to M is 2p = 2(m − r + 1). In the same way, the primary obstruction
dimension to the construction of an r-frame tangent to the 2s-dimensional stratum Vi is
2(s − r + 1). That means that if one intends to construct a stratified r-frame tangent to X
using the triangulation (K), then one will obtain obstruction cocycles on the strata with
different dimensions according to the dimension of the considered stratum. Considering
the triangulation (K), one cannot obtain a global cocycle with a well-defined dimension.
The M.-H. Schwartz observation is the following: Let us denote by (D) the dual
cell decomposition of (K) associated to a barycentric subdivision (K 0 ) (see 1.9). Each
(D)-cell is transverse to the strata. In particular, if d is a (D)-cell whose dimension
2p = 2(m − r + 1) is the obstruction dimension for the construction of an r-frame tangent
to M and if Vi is a stratum of dimension 2s, then the dimension of the cell d ∩ Vi is
dim(d ∩ Vi ) = 2(m − r + 1) − 2(m − s) = 2(s − r + 1)
that is precisely the obstruction dimension for the construction of an r-frame tangent to
Vi .
This observation leads naturally to the construction of a stratified vector field by
induction on the dimension of the strata, using the dual cell decomposition (D) and not
the triangulation (K).
The second observation by M.-H. Schwartz is that one has to consider stratified vector
fields and frames which are radial in the sense we explained in the previous section. Below,
53
we provide an explicit construction of radial extension of frames.
The obstruction dimension for a r-frame, over M is equal to 2p = 2(m − r + 1). That
means that one can construct such a section, without singularity over the (2p−1)-skeleton
D(2p−1) of D and with isolated singularities over D(2p) .
The obstruction dimension for an r-frame tangent to Vi2s is equal to 2q = 2(s − r + 1).
That means that one considers strata such that s ≥ r − 1. As we know, the dual cell
decomposition is transverse to the stratification, that means that if d2p is a 2p-dimensional
cell in D which intersects Vi2s , then d2p ∩Vi2s is a 2q-dimensional ∆ complex. Let us denote
∆2q = d2p ∩ Vi2s .
Let us consider a stratified r-frame v (r) = (v (r−1) , vr ), section of E r over ∆2q ⊂ Vi2s ,
with isolated singularities which are zeroes of the last vector vr . One suppose that vr has
length less than 1. One can define in a tube Tε (∆2q ) of radius ε (see above) on one hand
the parallel extension v̂ (r) = (v̂ (r−1) , v̂r ) of v (r) and on the other hand the radial transverse
vector field τ and one has:
Proposition 6.1 (Radial extension for a frame) If µ and ε are sufficiently small, the
radial extension of v (r) , defined by ṽ (r) = (v̂ (r−1) , v̂r + τ ) satisfies the following conditions:
1. the radial extension of vr satisfies the Proposition 5.9,
2. if the (r − 1)-frame v (r−1) has no singularity on ∆2q and if v (r) admits an isolated
singularity in a ∈ ∆2q ∩ K ⊂ Vi2s which is a zero of vr , then ṽ (r) = (v̂ (r−1) , v̂r + τ )
satisfies the same properties in Tε (∆2q ).
In that case, if the (r−1)-complex plane generated by v (r−1) (a) is linearly independent
of the tangent plane T (∆2q , a) in T (Vi2s , a), then the index of the extension ṽ (r) in a,
considered as an r-frame tangent to M is equal to the index of v (r) in a considered
as an r-frame tangent to Vi2s .
3. In the same hypothesis than (ii), if q = 0 (i.e; s = r − 1), and if a = ∆0 ⊂ Vi2s is a
zero of vr , then the index of v (r) in a is +1.
We will denote by I(v (r) , a) the index of v (r) in the isolated singularity a.
6.1.1
Global radial extension
The “global” construction of vector fields by radial extension goes as follows: If the lowest
dimensional stratum is a 0-dimensional one, i.e. a set of finitely many points ak , then
one consider a radial vector field v in a ball B(ak ) centered in each of these points. If the
lowest dimensional of strata is a stratum Vi of dimension 2s > 0, then one construct a
vector field v on Vi with isolated singularities. We notice that Vi is a manifold and it has
to be compact if X is compact; in this case the total Poincaré-Hopf index of v on Vi is
χ(Vi ).
Now we go up by increasing dimensions of the strata containing Vi in their closure:
if Vi ⊂ V j , then we extend v to a neighborhood of Vi in X as above, and then extend
it further to all of Vj with isolated singularities. We proceed further in this way to get
a stratified vector field v on all of X. Furthermore, by construction one has that if
54
we consider the closed strata V j , then the vector field is pointing inwards Vj near its
boundary. Thus one has, by theorem , that the Poincaré-Hopf index of v on each stratum
Vj is χ(Vj ).
2p
2s
One will construct v (r) over the subsets A2q
i = d ∩ Vi , by increasing dimensions of
the strata Vi . One will construct at each step over Ai and a tube Tε (Ai ), neighbourhood
of Ai in D(2p−1) . At the following steps, the vector field could be modified, but without a
tube Tε0 (Ai ) ⊂ Tε (Ai ).
i) If Vi2r−2 is a stratum whose real dimension is 2r − 2 = 2(m − p), the obstruction
dimension to the construction of a section of Vr (T Vi ) is zero. One takes any (r − 1)-frame
v (r−1) tangent to Vi2r−2 in the vertices aj = ∆0j of ∆ located in d2p ∩ Vi2r−2 and vr zero in
these points.
One construct the radial extension of the r-vector in the tubes Tε (∆0j ) as a r-frame
still denoted by v (r) . According to Proposition 6.1 (iii), one has I(v (r) , aj ) = +1.
ii) Let us suppose s > r − 1 and the construction already performed on all strata Vk
whose dimension is less than 2s. That means that the construction has been performed
on the sets Ak and the tubes Tε (Ak ). Let us consider a 2s-dimensional stratum Vi .
Let us denote Ȧi = Ai ∩ (Vi \ Vi ). The r-frame is already constructed over Ai ∩
Tε (Ȧi ). One can extend it on the rest of Ai with isolated singularities denoted by ak ans
located in 2q-open cells ∆2q
k located outside T1 (Ȧi ) (see[Sc4]) and such that, over a ball
bk neighbourhood of ak in ∆2q
k , one has:
a) v (r−1) (x) generates a (r − 1)-complex plane P 2r−2 (x) supplementary of T (∆2q
k , x) in
2s
T (Vi , x),
b) for x ∈ Bk \ {ak }, one has vr (x) tangent to ∆2q
k and it has length less than 1.
(r)
At that stage, one has v on Ai ∪ Tε (Ȧi ). One can extend it in a tube Tε0 (Ai ) (with
0
ε < ε1 < ε), in such a way that one has:
• the r-frame does not change within a tube Tε1 (Ȧi ) ⊂ Tε (Ȧi ),
• in Tε0 (Ai ) ⊂ Tε (Ȧi ), the extension is the radial extension of v (r) ,
• in Tε (Ȧi ) ⊂ Tε1 (Ȧi ), the obtained r-frame is linear combination of the r-frame
v (r) |Tε (Ȧi ) already constructed and of the radial extension of v (r) |Ai .
The constructed r-frame satisfies the properties of
Theorem 6.2 [B-S], [Sc2], [Sc5] One can construct, on the 2p-skeleton (D)2p , a stratified
r-frame v (r) , called radial frame, whose singularities satisfy the following properties:
(i) v (r) has only isolated singular points, which are zeroes of the last vector vr . On
(D)2p−1 , the r-frame v (r) has no singular point and on (D)2p the (r − 1)-frame v (r−1) has
no singular point.
(ii) Let a ∈ Vi ∩ (D)2p be a singular point of v (r) in the 2s-dimensional stratum Vi .
If s > r − 1, the index of v (r) at a, denoted by I(v (r) , a), is the same as the index of the
restriction of v (r) to Vi ∩ (D)2p considered as an r-frame tangent to Vi . If s = r − 1, then
I(v (r) , a) = +1.
(iii) Inside a 2p-cell d which meets several strata, the only singularities of v (r) are
inside the lowest dimensional one (in fact located in the barycenter of d).
55
(iv) The r-frame v (r) is pointing outwards a (particular) regular neighborhood U of X
in M . It has no singularity on ∂U .
6.2
Schwartz classes
Let us denote by T the tubular neighborhood of X in M consisting of the (D)-cells which
meet X. Let us recall that d∗ (σ) is the elementary (D)-cochain whose value is 1 at d(σ)
and 0 at all other cells. We can define a 2p-dimensional (D)-cochain in C 2p (T , ∂T ) by:
X
c̃ =
I(v (r) , σ̂) d∗ (σ).
d(σ)∈T
dim d(σ)=2p
In other words, the cochain c̃ satisfies
hc̃ · d(σ)i = I(v (r) , , σ̂).
Figure 13: The tubular neighborhood T
In a classical way the cochain is a cocycle, the obstruction cocycle (see [Sc4]) whose
class cp (X) lies in
H 2p (T , ∂T ) ∼
= H 2p (T , T \ X) ∼
= H 2p (M, M \ X),
where the first isomorphism is given by retraction along the rays of T and the second by
excision (by M \ T ).
56
Definition 6.3 [Sc2],[Sc5] The p-th Schwartz class is the class
cp (X) ∈ H 2p (M, M \ X).
M.-H. Schwartz proved that the class does not depend of the different choices involved in
its construction. The proof of this fact is now easier, using Theorem 6.24 below.
6.3
Nash transformation
Let M be an complex analytic manifold, of complex dimension m. Let X be a ndimensional subanalytic complex variety, X ⊂ M . Let us denote by Σ = Xsing the
singular part of X and by Xreg = X \ Σ its regular part.
The Grassmanian manifold of complex n-planes in Cm is denoted by Gn (Cm ). Let us
consider the Grassmann bundle of n (complex) planes in T M , denoted by G. The fibre
Gx over x ∈ M is the set of n-planes in Tx (M ), isomorphic to Gn (Cm ). An element of G
is denoted by (x, P ) where x ∈ M and P ∈ Gx .
On the regular part of X, one can define the Gauss map γ : Xreg −→ G by
γ(x) = (x, Tx (Xreg )).
e is defined as the closure of the image of γ
Definition 6.4 The Nash transformation X
in G.
Xreg
e = Imγ ,→ G
X
ν↓
↓
X
,→ M
G
%γ ↓
,→ M
(6.5)
Figure 14: The Nash transformation of the cone
e is not smooth, nevertheless, it is an analytic variety and the restriction
In general, X
e → X of the bundle projection G → M is analytic.
ν:X
57
Let us denote by E the tautological bundle over G. The fibre EP at a point (x, P ) ∈ G
is the set of the vectors v in the n-plane P ∈ Gx .
EP = {v(x) ∈ Tx M : v(x) ∈ P }
e
e = E| , then E|
e
Let us define E
X̃
X̃reg can be identified with T (Xreg ) where Xreg =
ν −1 (Xreg ) ∼
= Xreg and
e = E ×G X
e = {(v(x), x̃) ∈ E × X
e : v(x) ∈ x̃}
E
e is a n-complex plane in Tx (M ) and x = ν(x̃).
x̃ ∈ X
One has a diagram:
e −−−→ E
E




y
y
e −−−→
X


νy
G


y
X −−−→ M
e is written (x, P, v) where x ∈ X, P is a n-plane in ν −1 (x) and v is a
An element in E
vector in P . If x ∈ Xreg , then P = Tx (Xreg ).
Let us denote by {Vi } a complex analytic stratification of (M, X) satisfying the Whitney conditions.
The following lemma is fundamental for the understanding of the geometrical definition
of the local Euler obstruction. We recall the proof which is a direct application of the
Whitney condition (a).
Lemma 6.6 ([B-S, Proposition 9.1]) A stratified vector field v on A ⊂ X admits a canone
ical lifting ṽ on ν −1 (A) as a section of E.
ν∗
e −→
E
T M |X
ṽ ↑↓
v ↑↓
ν
e
X −→
X
ν∗ (x, x̃, v(x)) = (x, v(x)).
e Let
Proof:
Let us consider a stratified vector field v on A ⊂ X and a point x
e ∈ X.
us denote x = ν(e
x).
(i) If x ∈ Xreg , then v(x) ∈ Tx (Xreg ) = x
e with x
e = ν −1 (x). We define ṽ(x̃) =
(x, Tx (Xreg ), v(x)).
(ii) If x ∈ Vi , then v(x) ∈ Tx (Vi ). Each x̃ ∈ ν −1 (x) is in the closure of the image of
ereg such that x̃ = lim x
γ, i.e. there is a sequence (e
xn ) of points of X
fn , ν(f
xn ) = xn ∈ Xreg
and x
fn = Txn (Wreg ). Then one has lim(xn ) = x and lim Txn (Xreg ) = x̃. By the Whitney
condition (a), one has Tx (Vi ) ⊂ x̃ and we can define ṽ(x̃) = (x, x̃, v(x)).
58
6.4
Mather classes
We introduce now two ingredients useful to define the MacPherson classes, namely the
Mather classes and the local Euler obstruction.
Mather classes are defined as Chern classes of the Nash bundle, more precisely:
Definition 6.7 Let X ⊂ M a singular algebraic complex subvariety of a complex algebraic manifold M . The Mather class of X is defined by:
e ∩ [X])
e
cM (X) = ν∗ (c∗ (E)
e denotes the usual (total) Chern class of the bundle E
e in H ∗ (X)
e and the
where c∗ (E)
e is the Poincaré duality homomorphism.
cap-product with [X]
Let us recall that in general, Poincaré homomorphism is not an isomorphism.
The Mather classes do not satisfy the Deligne-Grothendieck’s conjecture that we state
later on (see Theorem 6.13).
6.5
Euler local obstruction
The notion of local Euler obstruction was defined originally by R. MacPherson [MP] in
1974. Definitions equivalent to MacPherson’s have been given by several authors. It
has been shown in [BDK] that the local invariant of singularities which appear in the
Kashiwara formula for the index of holonomic modules [Ka] is equal to the local Euler
obstruction.
The original definition, due to R. MacPherson [MP] uses differential form. We will
introduce the dual definition, using vector fields.
Notation Let us consider a bundle E over X ⊂ Cm and A ⊂ X with boundary ∂A. Let
us suppose that s is a section of E defined over the ∂A. The obstruction cocycle to extend
the section s inside A will be denoted by Obs(s, E, A).
Let z = (z1 , . . . , zm ) be local coordinates in Cm around {0}, such that zi (0) = 0, we
denote by Bε and Sε the ball and the sphere centered in {0} with radius ε in Cm . Let us
denote by Oν −1 (Bε ),ν −1 (Sε ) the orientation class (fundamental class)
Oν −1 (Bε ),ν −1 (Sε ) ∈ H2n (ν −1 (Bε ), ν −1 (Sε ); Z).
Let us recall that a radial vector field v in a neighborhood of the point {0} ∈ X is a
stratified vector field so that there exists ε0 > 0 such that for all ε, 0 < ε < ε0 , the vector
v(x) is pointing outwards the ball Bε over the boundary Sε = ∂Bε . By the Bertini-Sard
theorem, Sε is transverse to the strata Vi if ε is small enough, so the definition takes sense.
Definition 6.8 [B-S] Let v be a stratified radial vector field over X ∩ Sε and ṽ the lifting
of v on ν −1 (X ∩ Sε ). The local Euler obstruction Eu0 (X) is the obstruction to extend
ṽ as a nowhere zero section of Ẽ over ν −1 (X ∩ Bε ), evaluated on the orientation class
Oν −1 (Bε ),ν −1 (Sε ) :
e ν −1 (X ∩ Bε )).
Eu0 (X) = Obs(ṽ, E,
The local Euler obstruction is independent of all choices involved.
59
Figure 15: Local Euler obstruction
6.5.1
Properties of Euler local obstruction
Theorem 6.9 [MP],[GS] The Euler obstruction satisfies:
1. Eux (X) = 1 if x is a regular point of X;
2. Eux×x0 (X × X 0 ) = Eux (X) · Eux0 (X 0 );
3. If X is a curve, then Eux (X) is the multiplicity of X at x;
If X is the cone over a non singular plane curve of degree d and x is the vertex of
the cone, then Eux (X) = 2d − d2 ;
4. P
If X is locally reducible at x and Xi are its irreducible components, then Eux (X) =
Eux (Xi ).
The following proposition has been proved by many authors, in particular see [B-S] :
Proposition 6.10 (Constructibility) ([MP],[B-S],[Du] and other authors): The local Euler obstruction is constant along the strata of a Whitney stratification of X.
The main property of local Euler obstruction is the following Brasselet-Schwartz Proportionality Theorem [B-S]. Things being local, one can suppose we are in a ball Bε in
Cm local chart of M , with center a ∈ Vi ⊂ X.
Theorem 6.11 ([B-S], Théorème 11.1) (Proportionality Theorem for vector fields). Let
v be a stratified vector field obtained by radial extension (of a vector field on Vi ) with an
isolated singularity in the point a ∈ Vi , with index I(v, a) = I(v|Vi , a), then
e ν −1 (Bε ∩ X)) = Eua (X) · I(v, a).
Obs(ṽ, E,
e in an obvious way. We will denote by E
e r the
One defines the bundle of frames in E
e
bundle of r-frames in E.
60
Theorem 6.12 ([B-S], Théorème 11.1) (Proportionality Theorem for frames). Let v r be
a radial r-frame with an isolated singularities at the barycenter of the 2p-cells d2p = d(σ)
with index I(v r , σ̂) at {σ̂} = d2p ∩ σ. Then the obstruction to the extension of ṽ r as a
e r on ν −1 (d2p ∩ X) is
section of E
e r , ν −1 (d2p ∩ X)) = Euσ̂ (X) · I(v r , σ̂).
Obs(ṽ r , E
6.6
MacPherson classes
Let us recall firstly some basic definitions.
A constructible set in a variety X is a subset obtained by finitely many unions, intersections and complements of subvarieties. A constructible function α : X → Z is a
function such that α−1 (n) is a constructible set for all n. The constructible functions
on X form a group denoted by F(X). If A ⊂ X is a subvariety, we denote by 1A the
characteristic function whose value is 1 over A and 0 elsewhere.
If X is triangulable, α is a constructible function if and only if there is a triangulation (K) of X such that α is constant on the interior of each simplex of (K). Such a
triangulation of X is called α-adapted.
The correspondence F : X → F(X) defines a contravariant functor when considering
the usual pull-back f ∗ : F(Y ) → F(X) for a morphism f : X → Y . One interesting fact
is that it can be made a covariant functor when considering the pushforward defined on
characteristic functions by:
f∗ (1A )(y) = χ(f −1 (y) ∩ A),
y∈Y
for a morphism f : X → Y , and linearly extended to elements of F(X). The following
result was conjectured by Deligne and Grothendieck in 1969.
Theorem 6.13 [MP] Let F be the covariant functor of constructible functions and let
H∗ ( ; Z) be the usual covariant Z-homology functor. Then there exists a unique natural
transformation
c∗ : F → H∗ ( ; Z)
satisfying c∗ (1X ) = c∗ (X) ∩ [X] if X is a manifold.
The theorem means that for every algebraic complex variety, one has a natural transformation c∗ : F(X) → H∗ (X; Z) satisfying the following properties:
1. c∗ (α + β) = c∗ (α) + c∗ (β) for α and β in F(X),
2. c∗ (f∗ α) = f∗ (c∗ (α)) for f : X → Y morphism of algebraic varieties and α ∈ F(X),
3. c∗ (1X ) = c∗ (X) ∩ [X] if X is a manifold.
The MacPherson’s construction uses both the constructions of Mather classes and
local Euler obstruction.
61
Proposition 6.14 There is a isomorphism T between algebraic cycles on X and constructible functions, given by
X
X
T(
ni Vi )(p) =
ni Eup (Vi )
For a Whitney stratification, we have the following lemma:
Lemma 6.15 [MP] There are integers nα such that, for every point x ∈ X, we have:
X
nα Eux (Vα ) = 1.
α
Proof:
The proof is an easy exercise.
Definition 6.16 [MP] The MacPherson class of X is defined by
X
nα i∗ cM (Vα )
c∗ (X) = c∗ (1X ) =
α
where i denotes the inclusion Vα ,→ X.
Note that we have the following relation : cM (X) = c∗ (EuX ).
In particular, the Chern classes of an algebraic variety are represented by algebraic
cycles.
6.7
Schwartz and MacPherson classes
In this section, we show the Theorem 6.24 that states that Schwartz and MacPherson
classes correspond via Alexander duality (see section 5.2.1).
Proposition 6.17 The are a simplicial triangulation K of M compatible with the stratification and a cellular decomposition K̃ of the Grassman bundle G compatible with the
strata ν −1 (Vi ) such that:
1. equipped with the triangulation K (resp. with the decomposition K̃), M (resp. G)
is a combinatorial variety,
2. the triangulation of M and decomposition of G are C 1 -differentiable,
3. for each cell σ̃β , then ν(σ̃β ) is a simplex σα ,
4. the restriction of ν to each cell σ̃β has constant rank.
One says that the K̃-cell σ̃β is horizontal if one has dim ν(σ̃β ) = dim σ̃β . In that case,
the restriction of ν on the (open) cell σ̃β is a diffeomorphism on ν(σ̃β ).
˜ of K̃. Image of a ∆-simplex
˜
In the sequel, one defines a simplicial subdivision ∆
will
not be a ∆-simplex but will satisfy a property that is given in the following Proposition
(see [B-S]).
62
˜ of K̃ satisfying the
Proposition 6.18 . One can construct a simplicial subdivision ∆
following conditions:
˜
• (i) Every open cell σ̃β contains one and only one ∆-vertex
denoted by ãβ . Let aα be
the barycenter of σα = ν(σ̃β ), then ãβ is contained in the inverse image ν −1 (aα ),
˜
• (ii) Let σ̃β be a ∆-cell
with image σα , and σαi a simplex in the boundary of σα . Let
˜
us denote by d(σαi ) the dual cell of σαi , then σ̃β ∩ ν −1 (σαi ) is a ∆-complex.
˜ one has the following results:
From the construction of ∆,
Proposition 6.19 Let σαm a m-dimensional K-simplex, d(σαm ) the dual cell in M . Let
us denote by σ̃βm the horizontal K̃-cells whose image is σαm and d(σ̃βh ) their dual cells in
G. Then one has
[
Closure of ν −1 (d(σαm ) ∩ X) = Closure of
d(σ̃βh ) ∩ X̃.
(6.20)
β
Corollary 6.21 For each q-dimensional cell d(σα ), one has ν −1 (d(σα )) is a q-dimensional
˜
∆-complex.
Let σα2r−2 be a (2r − 2)-dimensional K-simplex, d(σα2r−2 ) its dual cell whose intersection with X is 2p-dimensional. It results from Corollary 6.21 that ν −1 (d(σα )) is a
˜
2p-dimensional ∆-complex.
One can proves the following Theorem:
˜
Theorem 6.22 Let c̃ be a ∆-cocycle
of the 2p-Chern class cp (Ẽ) of the Nash bundle Ẽ.
Let us denote kα = hc̃, ν −1 (Dα2p )i. Then the Mather class cM,r−1 (X) contains the cycle
X
kα σα2r−2
2r−2
σα
⊂X
˜
Proof: Let us consider the ∆-cycle
w̃2n which is the sum of all simplexes δ̃ 2n with the
orientation induced by the one of X̃. The class of w̃2n in H2n (X̃) is the fundamental class
2p
p
[X̃]. Let c̃ be a cocycle in C∆
˜ (X̃) representing the Chern class c (Ẽ), then a cycle of the
Mather class cM,r−1 (X) = ν∗ (cp (Ẽ) ∩ [X̃]) is given by
ν∗ (c̃ ∩ w̃2n ).
(6.23)
2p
Let us recall the result in [B1]: The Poincaré morphism C∆
˜ (X̃) → C2r−2,K̃ (X̃), capproduct by w̃2n is composed of the Alexander isomorphism ([B1], §3) and the Thom
˜ 2r−2 ) is the
homomorphism, dual of intersection with X̃. In another words, if d˜2t
β = d(σ̃β
dual cell, in G of the (2r − 2)-cell σ̃β2r−2 , then one has
c̃ ∩ w̃2n =
X
where νβ = hc̃ · d˜2t
β ∩ X̃i.
νβ σ̃β2r−2
σ̃β2r−2 ⊂X̃
(see [B1], formulae (7), (8) and diagram (16)).
63
Taking the image by ν∗ of that cycle, the contributions of horizontal cells σ̃β2r−2 are
the only one which do not vanish, the others having an image whose dimension is less
than 2r − 2. The cycle 6.23 is homologous to the cycle
X
X
ν∗ (
νβ σ̃β2r−2 ) =
kα σα2r−2
β
2r−2
⊂X
σα
P
P
2r−2
where kα = νβ = hc̃ · d˜2t
β ∩ X̃i, the sum being extended on indices β such that σ̃β
is horizontal with image σα2r−2 . One obtains the result, using Proposition 6.19.
The following result has been proved in [B-S]:
Theorem 6.24 [B-S] The MacPherson class is the image of the Schwartz class by the
Alexander duality isomorphism
∼
=
H 2(m−r+1) (M, M \ X) −→ H2(r−1) (X).
Proof: Using the notations of section 6.2, the r-frame v r determines a cocycle of the
M.H. Schwartz class:
X
X
∗
b
c=
να (d2q
with
να =
I(v r , aj ).
(6.25)
α )
d2q
α ∩X6=∅
aj ∈d2q
α ∩X
It determines also a cocycle c̃ of the Chern class cp (Ẽ) such that
< c̃.ν −1 (d2q
α ∩ X) >= Euaα (X)µα ,
2q
2r−2
where aα is any point of σα2r−2 , simplex whose the cell d2q
).
α is dual, i.e. dα = d(σα
M
Theorem 6.22 implies that the Chern-Mather class cr−1 (X) is represented by the cocycle:
X
Euaα (X)µα σα2r−2
2r−2
σα
⊂X
2r−2
where µα is the coefficient of the cocycle ĉ relatively to the cell d2q
and
α dual of σα
2r−2
where aα is any point of σα .
Let us write the previous result for each V i : The Chern-Mather class cM
r−1 (V i ) is
represented by the cocycle:
X
Euaα (V i )µα σα2r−2 .
2r−2
σα
⊂V i
By definition 6.16, the MacPherson class c∗ (X) is represented by the cocycle:
X
X
ni
Euaα (V i )µα σα2r−2 .
i
2r−2
σα
⊂V i
In this expression, the coefficient of µα σα2r−2 is
X
Cα =
ni Euaα (V i ) = 1, with Iα = {i : σα2r−2 ⊂ V i } = {i : aα ∈ V i }.
i∈Iα
64
One obtains a cycle of the MacPherson class of X of the form:
X
γ=
µα σα2r−2 .
2r−2
σα
⊂X
Let us recall (5.2.1) that the Alexander isomorphism H 2q (M, M − X) → H2r−2 (X) is
induced by the isomorphism:
2q
C(D)
(M, M − T̊ ) → C2r−2,(K) (X)
2r−2
∗
2q
such
which associates to a (D)-cochain (d2q
α ) such that dα ∩ X 6= ∅ the (K)-chain σα
2q
2r−2
that dα = d(σα ). By this isomorphism, the cycle γ is image of the cocycle of the
M.H.Schwartz class (cf 6.25), which proves the theorem.
We observe that we determined a cocycle of the MacPherson class. In fact, one has
the following corollary:
Corollary 6.26 Let (K) be a simplicial triangulation of M compatible with a Whitney
stratification and v r a r-radial frame defined on, the 2q-squeleton D(2q) of a cellular decomposition (D) dual of (K). The (r − 1)-st MacPherson class cr−1 (X) is represented by
the cycle
X
I(v (r) , σ̂) σ
σ∈X
where dim σ = 2(r − 1).
6.8
Schwartz-MacPherson classes for projective cones
In this section, we show the following result, due to Barthel, Brasselet and Fieseler [BBF]
following ideas of Brasselet and Gonzalez-Sprinberg [BG].
Theorem 6.27 Let Y ⊂ PN , be a projective variety and ı : Y ,→ KY the canonical
inclusion in the projective cone KY on Y with vertex {s}. Let us denote also by K :
H∗ (Y ) → H∗+2 (KY ) the homological projective cone, one has
cj (KY ) = ı∗ cj (Y ) + Kcj−1 (Y ),
(6.28)
where Kc−1 (Y ) denotes the class [s] ∈ H0 (KY ).
Let us consider an n-dimensional projective variety Y in PN and let us denote by L
the restriction of the hyperplane bundle of PN to Y .
If H denotes an hyperplane in PN , then D = Y ∩ H is a divisor in Y and we denote
by [D] ∈ H2n−2 (Y ) the fundamental class. By Poincaré duality, let ηD ∈ H 2 (Y ) be the
associated cohomology class. L is the bundle associated to D and one has c1 (L) = ηD .
We denote by E the completed projective space of the total space of L, i.e. P(L ⊕ 1Y )
where 1Y is the trivial bundle of complex rank 1 on Y . The canonical projection p : E → Y
admits two sections, zero and infinite, with images Y(0) and Y(∞) . The projective cone
KY is obtained as a quotient of E by contraction of Y(∞) in a point {s}. It is the Thom
space associated to the bundle L, with basis Y .
65
Let us consider p : E → Y as a sphere bundle with fibre S 2 , subbundle of a bundle
p̄ : Ē → Y with fibre the ball B 3 . We denote by θĒ ∈ H 3 (Ē, E) the associated Thom
class; One has a Gysin exact sequence
γ
pj
. . . → Hj+1 (Y ) → Hj−2 (Y ) → Hj (E) → Hj (Y ) → . . . ;
in which the gysin map γ is the composition of
(p̄j−2 )−1
(∩θ )−1
∂
Ē
Hj−2 (Y ) −→ Hj−2 (Ē) −→
Hj+1 (Ē, E) → Hj (E)
and can be explicited in the following way: If ζ is a cycle representing the class [ζ] of
Hj−2 (Y ), then γ([ζ]) is the class of the cycle p−1 (ζ) in Hj (E).
Let π the canonical projection π : E → KY .
Definition 6.29 We call homological projective cone and we denote by κ the composition
κ = π∗ γ : Hj−2 (Y ) → Hj (KY ) for j ≥ 2. We let κ(0) := [s] ∈ H0 (KY ) for 0 = H−2 (Y ).
Let us remark that for j 6= 0, κ is an homomorphism.
The theorem 6.27 is a direct consequence of the following proposition proved in [BBF].
Proposition 6.30 The Chern classes of E and Y are related by the formula
c∗ (E) = (1 + η0 + η∞ ) ∩ γ(c∗ (Y )),
(6.31)
where ηj := c1 O(Y(j) ) ∈ H 2 (E) for j = 0, ∞, and ∩ denotes the usual cap-product.
Proof of the Theorem 6.27 (from the Proposition 6.30). Let 1E the constructible function which is the characteristic function of E, then one has
(
χ(Y ), if x = s
π∗ (1E )(x) =
1,
elsewhere,
i.e.
π∗ (1E ) = 1KY + (χ(Y ) − 1)1{s} .
As one has
π∗ c∗ (1E ) = c∗ (π∗ (1E ))
one obtains
π∗ c∗ (E) = c∗ (KY ) + (χ(Y ) − 1) [s].
(6.32)
On another hand, from the formula (6.31) one obtains:
π∗ c∗ (E) = π∗ γ(c∗−1 (Y )) + π∗ (η0 ∩ γ(c∗ (Y ))) + π∗ (η∞ ∩ γ(c∗ (Y ))).
66
(6.33)
Let ι0 : Y ,→ E and ι∞ : Y ,→ E be the inclusions of Y as zero and infinite sections
of E respectively. By definition of γ, for a cycle ζ in Y and for j = 0 or ∞, one has
ηj ∩ γ([ζ]) = (ιj )∗ ([ζ])
then
π∗ (ηj ∩ γ(c∗ (Y ))) = π∗ ιj∗ c∗ (1Y ) = π∗ c∗ (1Y(j) ) = c∗ π∗ (1Y(j) ).
Let us denote by ι = π ◦ ι0 : Y ,→ KY the natural inclusion of Y in KY , one has
π∗ (1Y(0) ) = 1ι(Y ) and π∗ (1Y(∞) ) = χ(Y )1{s} .
One obtains
π∗ (η0 ∩ γ(c∗ (Y ))) = c∗ (1ι(Y ) ) = ι∗ c∗ (Y ),
and
π∗ (η∞ ∩ γ(c∗ (Y ))) = χ(Y )c∗ (1{s} ) = χ(Y )[s],
where [s] is the class of the vertex s in H0 (KY ). The comparison of the formulae (6.32)
and (6.33) gives:
c∗ (KY ) = ı∗ c∗ (Y ) + π∗ γc∗−1 (Y ) + [s],
and the Theorem 6.27.
6.9
Schwartz-MacPherson classes of Thom spaces associated to
embeddings
The previous construction associates canonically a Thom space to the embedding of a
smooth variety Y in PN . As examples, let us consider the image of the Segre embedding
P1 × P1 ,→ P3 , defined in homogeneous coordinates by
ϕS : (x0 : x1 ) × (y0 : y1 ) 7→ (x0 y0 : x0 y1 : x1 y0 : x1 y1 ),
That is an embedding whose bidegree is (1, 1). Image ϕS (P1 × P1 ) is a non degenerate
quadric Q provided with two families of generatices. Euler class of the bundle E in
H 2 (P1x × P1y ) = H 2 (P1x ) ⊕ H 2 (P1y ) = Z ⊕ Z is c1 (E) = (ηx , ηy ) where ηx is Euler class of
the hyperplane bundle of P1x , i.e. such that ηx ∩ [P1x ] = 1.
The image of the Veronese embedding P2 ,→ P5 defined by
ϕV : (x0 : x1 : x2 ) 7→ (x20 : x0 x1 : x0 x2 : x21 : x1 x2 : x22 ).
That is an embedding whose degree is 2. Image ϕV (P2 ) is smooth and has degree 4. It is
called Veronese surface. Euler class of the bundle E in H 2 (P2 ) is c1 (E) = 2ηK where H
being hyperplane in P5 , one has H ∩ V is a divisor in P2 ∼
= 2K, K being hyperplane in
2
1
2
2
P . Then ηK = c (EK ) is generator of H (P ), one has ηK ∩ [K] = 1.
With the previous construction, KY is the Thom space associated to the fibre bundle
L, of complex rank 1 and restriction to Y of the hyperplane bundle of PN . Chern classes
and intersection homology of these exemples have been computed in [BG]. In the case
67
of the Segre embedding, let d1 and d2 two fixed lines belonging each to a system of
generatrices of the quadric Y = P1 × P1 . Let us denote by ω the canonical generator of
H 2 (P1 ), one has c∗ (P1 ) = 1 + 2ω and c∗ (P1 ) = [P1 ] + 2??. Then
c∗ (Y ) = c∗ (P1 × P1 ) = ([Y ] + 2[d1 ]) ∗ ([Y ] + 2[d2 ]) = [Y ] + 2([d1 ] + [d2 ]) + 4[a]
where a is a point in Y and where ∗ denotes the intersection of cycles or homology classes.
One has
κ(c∗ (Y )) = [KY ] + 2([Kd1 ] + [Kd2 ]) + 4[Ka].
let us denote by ∼ the homology relation of cycles. In KY , one has [BG], §3:
Y ∼ Kd1 + Kd2 ,
d1 ∼ d2 ∼ Ka,
a ∼ s,
and, with 6.27
c∗ (KY ) = [KY ] + 3([Kd1 ] + [Kd2 ]) + 8[Ka] + 5[s] ,
| {z } |
{z
} | {z }
|{z}
H6 (KY )
H4 (KY )
H2 (KY )
H0 (KY )
which is the result of [BG].
In the case of the Veronese embedding, let d be a projective line in Y := P2 , one has:
∗
c (P2 ) = 1 + 3ω + 3ω 2 where ω is the canonical generator of H 2 (P2 ), and is dual, by
Poincaré isomorphism of the class [d] ∈ H2 (P2 ). One has, by Poincaré duality
c∗ (Y ) = [Y ] + 3[d] + 3[a]
where a is a point in Y . One has
K(c∗ (Y )) = [KY ] + 3[Kd] + 3[Ka]
such that, in KY , [BG], §3.b, Y ∼ 2Kd, d ∼ 2Ka and a ∼ s. One has
c∗ (KY ) = [KY ] + 5[Kd] + 9[Ka] + 4[s].
| {z } | {z } | {z }
|{z}
H6 (KY )
H4 (KY )
68
H2 (KY )
H0 (KY )
7
Other classes and comparisons
7.1
Fulton classes
The Fulton classes ([Fu] exemple 4.2.6 (a)) and Fulton-Johnson classes ([FJ], [Fu] exemple
4.2.6 (c)) have been defined in a general framework. In the case of Local Complete
Intersection, these classes coincide and can be defined in the following (simpler) way:
If X is a local complete intersection, then the normal bundle of Xreg in M extends
canonically to X as a vector bundle NX M and the Fulton class is equal to
cF (X) = c(T M |X )c(NX M )−1 ∩ [X] = c(τX ) ∩ [X].
Here τX = T M |X − NX M denotes the virtual tangent bundle on X, defined in the
Grothendieck group of vector bundles on X.
If X is a manifold, then cF (X) is the usual Chern class c∗ (X).
Let M be a non-singular compact complex analytic variety of pure dimension n + 1
and let L be a holomorphic line bundle on M . Take f ∈ H 0 (M, L), a holomorphic section
of L, such that the variety X of zeroes of f is a (nowhere dense) hypersurface in M . Then,
the Fulton class of X is
cF (X) = c(T M |X − L|X ) ∩ [X].
In [BLSS1], the authors show the following result:
Theorem 7.1 Let us assume that X ⊂ M is a hypersurface, defined by X = f −1 (0),
where f : M → D is a holomorphic function into an open disc around 0 in C. For
each point a ∈ X, let Fa denote a local Milnor fiber, and let χ(Fa ) be its Euler-Poincaré
J
characteristic. Then the Fulton-Johnson class cFr−1
(X) of X of degree (r−1) is represented
in H2(r−1) (X) by the cycle
X
χ(Fσ̂α ) I(v r , σ̂α ) · σα
(7.2)
σα ⊂X, dim σα =2(r−1)
7.2
Milnor classes
The comparison between the Schwartz-MacPherson classes and the Fulton-Johnson classes can be viewed in two ways, which coincide in some classical situations.
In the case of isolated singularities, the difference between Schwartz-MacPherson classes
and Fulton-Johnson classes is given by the Milnor numbers at the singular points:
69
Theorem 7.3 [SS] If X is compact and the singularities of X are isolated points {xi }
where X is a local complete intersection. Then
F
n+1
c∗ (X) − c (X) = (−1)
q
X
µxi [xi ] ∈ H0 (X).
i=1
This motivates the following definition given by various authors:
Definition 7.4 ([A1],[BLSS1],[PP2], [Yo]) The difference class
µ∗ (X) = (−1)n (cF (X) − c∗ (X))
is called the Milnor class of X.
7.2.1
Description in terms of constructible functions
The following description comes from [PP2].
Let us come back to the hypersurface situation: M is a non-singular compact complex
analytic variety of pure dimension n + 1 and L is a holomorphic line bundle on M . The
hypersurface X in M is the set of zeroes of a holomorphic section of L.
Consider the function χ : X → Z defined by χ(x) := χ(Fx ), where Fx denotes the
Milnor fibre at x and χ(Fx ) its Euler characteristic. Define also the function µ : X → Z
by µ = (−1)n−1 (χ − 1X ).
Fix any stratification S of X such that µ is constant on the strata of S, for instance
a Whitney stratification of X. The topological type of the Milnor fibre is constant along
the strata of the Whitney stratification of Z. Let us denote by µS the value of µ on the
stratum S.
Let
X
α(S 0 )
α(S) = µS −
S 0 6=S,S⊂S 0
be the numbers defined inductively on descending dimensions of S.
Theorem 7.5 [PP2] We have
µ∗ (X) =
X
α(S)c(L|X )−1 ∩ (iS,X )∗ c∗ (S) = c(L|X )−1 ∩ c∗ (µ),
S∈S
where iS,X : S → X denotes the natural inclusion.
The formula was conjectured in [Yo] when X is projective. Under this last assumption,
[PP1] proved earlier that
Z
Z
X
µ∗ (X) =
α(S) c(L|S )−1 ∩ c∗ (S)
X
S
S∈S
70
7.3
Motivic Chern classes: Hirzebruch theory for singular varieties
In the same way that the MacPherson Chern functor generalises the Chern class, the Todd
class and the Thom-Hirzebruch class have been generalised as natural transformations
respectively by Baum-Fulton-MacPherson and by Cappell-Shaneson. We show that the
motivic theory allows to unify the three generalisations in the case of singular varieties.
In the case of singular varieties, the three classes (Chern Todd and L-class) have been
generalized as natural transformations of functors, in the following way:
Definition 7.6 [Chern Transformation (MacPherson)] (see [MP])
Theorem 6.13 tells us that there is an unique natural transformation
c∗ : F(X) → H∗ (X)
from the group of constructible functions F(X) to homology, satisfying the suitable properties. In particular for the constructible function 1X , one defines c∗ (X) := c∗ (1X ) :
Schwartz-MacPherson class of X.
Definition 7.7 [Todd Transformation (Baum-Fulton-MacPherson)] [BFM]
There is an unique natural transformation
td∗ : G0 (X) → H∗ (X) ⊗ Q
from the Grothendieck group of coherent sheaves on X, satisfying suitable axioms, in
particular, for the structure sheaf OX on a smooth variety, td∗ (OX ) is the Todd class of
X. In general, one defines td∗ (X) := td∗ ([OX ]) as being the Baum-Fulton-MacPherson
Todd class of X.
Definition 7.8 [L-Transformation (Cappell-Shaneson)] [CS1, CS2]
There is an unique natural transformation
L∗ : Ω(X) → H2∗ (X; Q)
from the group of constructible self-dual sheaves on X, satisfying suitable axioms, in
particular, for the intersection sheaf IC X on a smooth variety, L∗ ([IC X ]) is the L-class of
X. In general, one defines L∗ (X) := L∗ ([IC X ]) as being the Cappell-Shaneson L-class of
X.
In short, one has the following table:
X manifold
number
χ(X)
—
χa (X)
—
sign(X)
X singular variety
cohomology classes
homology classes
Chern
Schwartz-MacPherson
—
—
Todd
Baum-Fulton-MacPherson
—
—
Thom-Hirzebruch
Cappell-Shaneson
71
The problem is that the three transformations are defined on different spaces:
G0 (X)
F(X),
and
Ω(X)
and one asks for the possibility of unifying in the same way thatn the Hirzebruch theory in
the smooth case. The problem has been solved (see [BSY]) using the motivic framework.
Let us recall some ingredients which will be useful:
Definition 7.9 Let X be an algebraic variety, the Grothendieck relative group of algebraic
varieties over X denoted by
K0 (var/X)
is the quotient of the free abelian group of isomorphy classes of algebraic maps Y −→ X,
modulo the “additivity relation”:
[Y −→ X] = [Z −→ Y −→ X] + [Y \ Z −→ Y −→ X]
for closed algebraic sub-spaces Z in Y .
In [BSY], the authors prove the following 4 theorems:
Theorem 7.10 The map
e : K0 (var/X) −→ F(X)
defined by
e([f : Y → X]) := f! 1Y
is the unique group morphism which commutes with direct images for proper maps and
such that
e([idX ]) = 1X for X smooth and pure dimensional.
Theorem 7.11 There is an unique group morphism
mC : K0 (var/X) −→ G0 (X)
which commutes with direct images for proper maps and such that
mC([idX ]) = [OX ]
for X smooth and pure dimensional.
Theorem 7.12 The morphism
sd : K0 (var/X) −→ Ω(X) defined by
sd([f : Y → X]) := [Rf∗ QY [dimC (Y )+dimC (X)]]
is the unique group morphism which commutes with direct images for proper maps and
such that
sd([idX ]) = [QX [2 dimC (X)]] = [IC X ]
for X smooth and pure dimensional.
72
Theorem 7.13 There is an unique group morphism
Ty : K0 (var/X) −→ H∗ (X) ⊗ Q[y]
which commutes with direct images for proper maps and such that
g
Ty ([idX ]) = td
(y) (T X) ∩ [X]
for X smooth and pure dimensional.
In particular, one has: T−1 ([idX ]) = c∗ (X).
Remark 7.14 If a complex algebraic variety X has only rational singularities (for example if X is a toric variety), then:
mC([idX ]) = [OX ] ∈ G0 (X) and in this case T0 ([idX ]) = td∗ (X).
That is not true in general !
The main result is the following:
Theorem 7.15 One has a commutative “tripode” diagramme:
F(X)
e
←−
mC
−→
K0 (var/X)
sd
c∗ ↓
G0 (X)
&
Ty ↓
↓ td∗
Ω(X)
y=−1
H∗ (X) ⊗ Q ←− H∗ (X) ⊗ Q[y]
L∗ ↓
y=0
−→
H∗ (X) ⊗ Q.
&
H∗ (X) ⊗ Q
y=1
7.4
Verdier Riemann-Roch Formula
Theorem 7.16 Let f : X 0 → X be a smooth map (or a map with constant relative
dimension), then one has
∗
∗
g
td
(y) (Tf ) ∩ f Ty ([Z −→ X]) = Ty f ([Z −→ X]).
Here Tf is the bundle over X 0 of tangent spaces to fibres of f .
Proposition 7.17 (Factorisation of Ty ) Let us define
P
−i
f
td(1+y) ([F]) := <∞
i=0 tdi ([F]) · (1 + y) . Then one has:
Ty = td(1+y) ◦ mC : K0 (var/X) −→ H∗ (X) ⊗ Q[y].
73
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