A NALYTIC K- HOMOLOGY AND BIRATIONAL GEOMETRY
Michel Hilsum, CNRS, Paris
Index Theory and its Ramifications in Geometry, Topology, and Physics
Fudan University, Shanghai, March 23-27, 2017
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B IRATIONAL EQUIVALENCE
- Algebraic projective variety = algebraic subset of PN (C) = zero set of a finite family of
homogeneous polynomials in C[Z0, Z1, · · · , ZN ]
It is not in general a manifold, but it is smooth on a Zariski open set.
- Morphism : Polynomial function V → W
- Rational map : φ : V → W partially defined morphism O ⊂ V → U ⊂ W
- Birational transformation : Rational map with a rational inverse
V, W smooth compact complex algebraic manifolds, ϕ birational equivalence = biholomorphic bijection between Zariski open subsets of V and W .
Exemple : Blow-up along an analytic φ : V̂ → V .
Remark : A birational φ : V − −− > W is a pair of birational morphisms.
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B IRATIONAL INVARIANTS ( NOT EXHAUSTIVE ...)
V, W birational
- OV and more generally H k (., OV ) for V smooth
- χ(V ) : complex Euler caracteristic
- Fundamental group :
V
ϕ@ f∗
@
R
@
-
Bπ
g∗
W
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Let V smooth projective, K0a(V ) = KK(C(V ), C) the analytic K-homology group and
DV ∈ K0a(V ) the K-homology class determined by the Dolbeault operator.
J. Rosenberg : f∗(DV ) = g∗(DW ) in K0(Bπ) ?
Positive answer with Strong Novikov conjecture : J. Rosenberg (higher index + rational
injectivity of assembly map)
Positive answer in general : J. Block - S. Weinberger :
- Topological K-homology (defined by Alexander duality)
- Relies on Grothendieck-Riemann-Roch theorem with value in topological K-homology
proved by Baum-Fulton-McPherson
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D IRECT PROOF IN ANALYTICAL K- HOMOLOGY ?
V, W smooth compact complex algebraic manifolds, ϕ birational equivalence, T topological space locally compact, f, g continuous maps, such that the diagram commutes :
V
ϕ@ f∗
@
R
@
-
W
g∗
T
Theorem (M.H.) : f∗(DV ) = g∗(DW ) in K0a(T )
Corollary : If ϕ is a birational morphism, then ϕ∗(DV ) = DW in K0a(W )
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Topological K-homology
Suppose V embedded continuously in S N .
K0t (V ) = K0(S N , S N − V )
Analytic K-homology
Bivariant K-theory
G. Kasparov’s bivariant K-theory : A, B C*-algebras
KK(A, B)
Cup product : KK(A, B) × KK(B, C) → KK(A, C)
Cycles : Bounded triples (E, F, τ ) or Unbounded triples (E, D, τ )
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Ka0(A) = KK(A, C)
When A = C(V ) commutative :
K0a(V ) = Ka0(C(V ))
Elliptic differential operators
They are isomorphic when V is a finite CW-complex.
V smooth compact complex hermitian manifold.
Dolbeault operator : ∂¯ : Cc∞(V ∞, Λ0,k ) → Cc∞(V ∞, Λ0,k+1)
Dirac-Dolbeault : DV = ∂¯ + ∂¯∗ selfadjoint on L2(V, Λ0,∗(V )).
Fundamental class : [DV ] ∈ K0a(V )
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Relative index theorem of Gromov-Lawson
K1 ⊂ V1 K2 ⊂ V2 are closed subset of spinc manifolds, ψ : V1 − K1 → V2 − K2
is a diffeomorphism preserving the spinc structures. Let Y ⊂ Vj − Kj be a common
hypersurface the complementary of Kj . Chop the manifolds at Y and glue them together
to obtain a manifold V . Let W be another manifold with boundary Y and glue it to Xj .
Then
Ind(D(V )) = Ind(D(W1)) − Ind(D(W2))
We want an anologous statement in K-homology
Next we suppose that ψ : V1 → V2 is everywhere defined and remains a diffeomophism
outside K1, and suppose that V1 and V2 are compact :
[D(V )] = [D(V1)] − ψ∗[D(W2)] in K0(V1)
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G ROMOV-L AWSON FOR FAMILIES
Next, suppose V1 and V2 are complex manifolds and that M is a complex manifold embedded in Vj and let Uj a tubular neighborhood , and ψ : V1 → V2 is holomorphic and
induces diffeo outside M . Let Lj be the family of Dirac-Dolbeault operators on the fiber
of Uj → M .
We can glue Uj together along they boundary and let L the family of glued operators.
Definition : Ind(L1, L2) = Ind(L) in K 0(M )
Thm : [D(V1)] − ψ∗[D(W2)] = j∗(Ind(L1, L2)) ∩ [DM ]) in K0(V1)
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Theorem : f∗(DV ) = g∗(DW ) in K0a(T )
It suffices to prove the theorem when ϕ : W → V is a morphism.
Suppose first that ϕ : W → V is the blow-up of V along a sub manifold M embedded
in V .
¯ cohomology is preserved by birational
The Ind(L1, L2)) = 0 essentially because the ∂equivalence.
In general we use a factorization theorem for birational morphism.
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FACTORIZATION OF BIRATIONAL MORPHISM (W ŁODARCZIK )
Let W → V be a birational morphism. There exists a sequence
s0
s1
sk−1
sk
Z0 99K Z1 99K Z2 · · · 99K Zk 99K Zk+1
sj
where Zj is an analytic complex manifolds, Z0 = V, Zk+1 = W and each symbole 99K
means either the canonical projection of a blow up with smooth center Zj+1 → Zj , either
the inverse of such Zj → Zj+1 and such that there exist tj : Zj → V with t0 = IdV ,
tk = ϕ and, either tj ◦ sj = tj+1, or tj = tj+1 ◦ sj .
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A NALYTIC K- HOMOLOGY OF ALGEBRAIC MANIFOLDS
V ⊂ PN (C) in general is not smooth.
Resolution of singularities p : V̂ → V
Corollary : The class p∗(DV ) ∈ K0(V ) is independent of the choice of V̂
Definition : Let Φµ(V ) ∈ K0a(V ) be the K-homology class obtained.
Dolbeaut operator : ∂¯ : Cc∞(V ∞, Λ0,k ) → Cc∞(V ∞, Λ0,k+1)
There is ∂¯min and ∂¯max.
There are different in general, even for complex algebraic curves : J. Brüning, N. Peyerimhoff, and H. Schröder (1990)
- Two natural selfadjoint extensions of D :
Dmin = ∂¯min − ∗∂¯max∗
DM = ∂¯max − ∗∂¯min∗ = − ∗ Dmin∗
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W HEN DOES Dmin
WITH COMPACT RESOLVENT ?
At the time being, only isolated singularities :
- D. Grieser & M. Lesch (1999), J. Ruppenthal (JFA, 2011) : ∂¯min has compact inverse
on its image.
However, if the singular set is smooth, then it remains true.
Whitney stratification.
T. Mostowski : Whitney stratification are Lipschitz stratification
Corollary : If the singular set is a manifold, then Dmin has compact resolvent
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In general, it is not known in general if compact resolvent.
The index of Dmin is finite , and is equal to the Euler characteristic of any smooth resolution (Pardon, Stern, Ruppenthal).
If it has compact resolvent, then it defines a class in K0(V ).
Suppose then it is so and let π : V̂ → V be a smooth resolution :
Theorem (M.H.) : π∗[D(V̂ )] = [D(V )] in K0a(V )
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Pardon, Stern, Ruppenthal : invariance of the sheaf of min L2 cohomology
Complex projective variety are topologically stratified manifold (Whitney)
T. Mostowski : Complex projective variety have Lipschitz stratifications.
It shows that along a strata, the family of Dirac operator is continuous and thus we get a
KK-cycle.
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L OCALE COMPLETE INTERSECTIONS
V ⊂ PN (C)
In general, Φµ(V ) cannot be seen as the fundamental class.
If V is lci, then there exists normal bundle N → V (which extends the normal bundle on
the regular part of V .
ch Φµ(V ) 6= Td(N ) ∩ [V ]
For example, if V is a curve then we have :
Td(N ) ∩ [V ] = ch Φµ(V ) + 12
where µP = multiplicity of P ∈ V .
P
P
µP (µP − 1)
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In general if F is a coherent sheaf on V , we may define Φµ(V, F) as follows :
Nash transform of F.
There exists a smooth resolution π : V̂ → V and a vector bundle E → V̂ such that
p∗(E) − F is supported on the complementary of an analytic subset Kn−1 of dimension
≤ n − 1 of V .
Such a resolution is unique up to a birational equivalence. Define
Φµ(V, F) = π∗([E] ∩ [DV ])
We thus get a stratification V0 ⊂ V1 ⊂ · · · ⊂ Vn = V and for each k a coherent sheaf Fk
on V̄k .
Proposition : If V is lci, then :
Td(N ) ∩ [V ] =
P
k (−1)
k
Φµ(Vk , Fk )
Remark : Thom-Mather characteristic class are defined analogously
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Happy Birthday !
Bon anniversaire Alain !