Parabolic bundles on curves
Parabolic bundles in higher dimension
Stacks of roots
Open problems
Parabolic vector bundles and
Nori’s fundamental group scheme
Niels Borne1
1 Université
Lille
Spring school on the fundamental group scheme, 6.5.2014
Niels Borne
Parabolic vector bundles
Parabolic bundles on curves
Parabolic bundles in higher dimension
Stacks of roots
Open problems
Plan
Parabolic bundles on curves
The original definition
Algebraic version of Mehta-Seshadri correspondence I
Finite parabolic bundles on the projective line
Parabolic bundles in higher dimension
Simpson’s definition
Biswas construction
Algebraic version of Mehta-Seshadri correspondence II
Stacks of roots
Parabolic bundles as orbifold bundles
Log geometry
Open problems
Parabolic bundles and connections
Wild parabolic bundles
Niels Borne
Parabolic vector bundles
Parabolic bundles on curves
Parabolic bundles in higher dimension
Stacks of roots
Open problems
The original definition
Algebraic version of Mehta-Seshadri correspondence I
Finite parabolic bundles on the projective line
The definition of Mehta-Seshadri
Definition (M-S 1980)
Let X be a projective, smooth curve over an algebraically
closed field k, D a reduced effective divisor on X .A parabolic
bundle E∗ = (E, F∗ , α∗ ) on (X , D) is the data of
1. a locally free sheaf E on X ,
2. for every point x in the support of D, of a filtration of the
residual stalk Ex := Ex ⊗OX ,x k (x):
Ex = F1 (Ex ) ⊃ F2 (Ex ) ⊃ · · · ⊃ Fnx (Ex ) ⊃ Fnx +1 (Ex ) = 0,
3. a sequence of real numbers (weights) (αx,i )1≤i≤nx verifying
0 ≤ αx,1 < · · · < αx,nx < 1.
If mx,i = dimk (x) (Fi (EP
(Ex )) (multiplicities), one sets
x )/Fi+1
P
x
degpar E∗ = deg E + x∈|D| ni=1
mx,i αx,i .
Niels Borne
Parabolic vector bundles
Parabolic bundles on curves
Parabolic bundles in higher dimension
Stacks of roots
Open problems
The original definition
Algebraic version of Mehta-Seshadri correspondence I
Finite parabolic bundles on the projective line
The correspondence
One assumes k = C, the genus of X is g ≥ 2.
Denote by U = X \D, and U an the (open) Riemann surface
associated to U.
Let V be a finite-dimensional C-vector space, and
ρ : π1top (U an , u) → U(V ) a unitary representation.
Out of this data, one constructs a parabolic bundle
(E∗ )ρ = (E, F∗ , α∗ )ρ on X (see later on for ρ of finite image).
One shows that (E∗ )ρ is (parabolic) semi-stable and that
degpar (E∗ )ρ = 0.
Theorem (Mehta-Seshadri 1980)
A parabolic bundle E∗ of parabolic degree 0 on (X , D) is
(parabolic) stable if and only if it arises from an irreducible
unitary representation ρ of π1top (U an , u).
Niels Borne
Parabolic vector bundles
Parabolic bundles on curves
Parabolic bundles in higher dimension
Stacks of roots
Open problems
The original definition
Algebraic version of Mehta-Seshadri correspondence I
Finite parabolic bundles on the projective line
Essentially finite parabolic bundles
Definition
Let X be a proper, reduced scheme over a field k .
1. (Weil) A locally free sheaf E is finite if there exists f 6= g in
N[t] so that f (E) ' g(E).
2. (Nori, B-V) E is essentially finite if it is the kernel of a
morphism between two finite locally free sheaves.
Theorem (tame version of M.Nori 1982)
Let X be a projective, smooth curve over an algebraically
closed field k, D a reduced effective divisor on X .
1. The category EF Par Vect(X , D) is tannakian.
2. Assume k is of characteristic 0. The Tannaka group of
EF Par Vect(X , D) is isomorphic to π1et (X \D, x).
Niels Borne
Parabolic vector bundles
Parabolic bundles on curves
Parabolic bundles in higher dimension
Stacks of roots
Open problems
The original definition
Algebraic version of Mehta-Seshadri correspondence I
Finite parabolic bundles on the projective line
The example of P1 \{0, 1, ∞}
Nori’s theorem gives an algebraic proof that representations of
π1et (P1 \{0, 1, ∞}, x) are in one to one correspondence with
finite parabolic bundles on (P1 , {0, 1, ∞}).
c2 .
Moreover, it is known via GAGA that π1et (P1 \{0, 1, ∞}, x) ' F
Problem
Prove algebraically (say, without GAGA) that finite parabolic
bundles on (P1 , {0, 1, ∞}) are in one to one correspondence
c2 .
with representations of F
Theorem (B. 2004)
Let (E, F∗ , α∗ ) be a finite parabolic bundle on (P1 , {0, 1, ∞}).
Then:
E ' OP1 (−2)⊕c ⊕ OP1 (−1)⊕b ⊕ OP⊕a
1
where a, b, c ∈ N determined by rk E, dim H 0 (X , E) and deg E.
Niels Borne
Parabolic vector bundles
Parabolic bundles on curves
Parabolic bundles in higher dimension
Stacks of roots
Open problems
Simpson’s definition
Biswas construction
Algebraic version of Mehta-Seshadri correspondence II
Parabolic bundles: reformulation of the definition
Let X be a scheme, D = (Di )i∈I a finite family of effective
Cartier divisors with normal crossings. Let r = (ri )i∈I be a family
of nonnegative integers.
Definition (C.Simpson around 1990)
Q
A parabolic bundle E· on (X , D) with weights in i∈I r1i Z is the
data
Q
1. For all m ∈ i∈I r1i Z of a locally free sheaf Em , verifying
Em0 ⊂ Em for m ≤ m0
Q
2. For m ∈ i∈I r1i Z, and n ∈ ZI , of compatible pseudo-period
isomorphisms
X
Em+n ' Em ⊗OX OX (−
ni Di )
i∈I
Niels Borne
Parabolic vector bundles
Parabolic bundles on curves
Parabolic bundles in higher dimension
Stacks of roots
Open problems
Simpson’s definition
Biswas construction
Algebraic version of Mehta-Seshadri correspondence II
Parabolic bundles and covers
Let X be a scheme, D a Cartier divisor.
Let π : Y → X a finite Galois cover of Galois group G and
tamely ramified along D, of ramification index r .
Let F be a G-sheaf locally free on Y (that is equipped with
isomorphisms (g ∗ F ' F)g∈G , verifying the cocycle condition).
To this data, I.Biswas associates the parabolic bundle E· on
(X , D) defined by:
Em = π∗G (OY ([−mπ ∗ D]) ⊗OY F)
Theorem (I.Biswas 1997)
The association F 7→ E· defines a one to one correspondence
between G-sheaves locally free on Y and parabolic sheaves on
(X , D) with weights in 1r Z, that preserves the tensor product.
Niels Borne
Parabolic vector bundles
Parabolic bundles on curves
Parabolic bundles in higher dimension
Stacks of roots
Open problems
Simpson’s definition
Biswas construction
Algebraic version of Mehta-Seshadri correspondence II
Nori’s theorem in higher dimension: snc divisor
Theorem (B.2009)
Let X be a proper, normal, connected scheme over an
algebraically closed field k , D a family of irreducible divisors
with simple normal crossings in X , D = ∪i∈I Di .
1. The category EF Par Vect(X , D) is tannakian.
2. Assume k is of characteristic 0. The Tannaka group of
EF Par Vect(X , D) is isomorphic to π1et (X \D, x).
Niels Borne
Parabolic vector bundles
Parabolic bundles on curves
Parabolic bundles in higher dimension
Stacks of roots
Open problems
Parabolic bundles as orbifold bundles
Log geometry
Quotient stack
Starting from a scheme Y endowed with the action of a group
G, one disposes of the quotient stack [Y /G].
Technically, this is the stack associated to the groupoid
pr2
G×Y ⇒Y .
a
Y
π
G−torsor
+
g [Y /G]
g
g
g
sg
X = Y /G moduli space
[Y /G] → Y /G is an isomorphism if and only if the action of G
on Y is free.
Sheaves on [Y /G] are in bijection with G-sheaves on Y .
Niels Borne
Parabolic vector bundles
Parabolic bundles on curves
Parabolic bundles in higher dimension
Stacks of roots
Open problems
Parabolic bundles as orbifold bundles
Log geometry
Stack of roots
Let X be a scheme, D an effective Cartier divisor, r ≥ 1.
If D is principal, defined by h = 0, one sets
p
r
D/X = X ×A1 [A1 /µr ], where X → A1 is given by h.
More generally, one shows that [A1 /Gm ] identifies with the
stack classifying invertible sheaves equipped with a section.
Definition (A.Vistoli 2000)
Given X un scheme, D an effective Cartier divisor, r ≥ 1, one
sets :
p
r
D/X = X ×[A1 /Gm ] [A1 /Gm ]
where the product is taken with respect to the morphims
(OX (D), sD ) : X → [A1 /Gm ] and ×r : [A1 /Gm ] → [A1 /Gm ].
Niels Borne
Parabolic vector bundles
Parabolic bundles on curves
Parabolic bundles in higher dimension
Stacks of roots
Open problems
Parabolic bundles as orbifold bundles
Log geometry
The correspondence
Theorem (B. 2007)
p
Let π : r D/X → X be the morphism
space, ∆
p to the moduli
∗ D = r ∆.
the canonical Cartier divisor on r D/X so that
π
p
The functor that to a locally free sheaf F on r D/X associates
the parabolic bundle with weights in 1r Z defined by
E l = π∗ (F ⊗ O(−l∆)), is an equivalence of tensor categories.
r
Niels Borne
Parabolic vector bundles
Parabolic bundles on curves
Parabolic bundles in higher dimension
Stacks of roots
Open problems
Parabolic bundles as orbifold bundles
Log geometry
Stack of roots are fine only in the snc case
Lemma (Variation on Abhyankar’s lemma)
Let X be a normal, connected scheme, D = (Di )i∈I a family of
irreducible divisors with simple normal crossings. Let Y → X a
cover tamely ramified along D, of ramification multi-index r, and
Galois with automorphism
group G.Then the natural morphism
p
r
of stacks [Y /G] → D/X is an isomorphism.
Remark
1. This is false for a general normal crossings divisor.
2. For instance if D is a self-crossing irreducible divisor,
[Y /G] is locally the stack classifying roots of OX (D) on
each branch, not only global roots.
3. Accordingly, the definition of parabolic bundles should be
modified and involve sheaves of weights.
Niels Borne
Parabolic vector bundles
Parabolic bundles on curves
Parabolic bundles in higher dimension
Stacks of roots
Open problems
Parabolic bundles as orbifold bundles
Log geometry
Sheaves of weights within log geometry
Definition (Kato around 1990)
A pre-log-scheme (X , α) is a scheme endowed with a
morphism of sheaves of monoids on Xet : α : M → (OX , ×) .
This is a log-scheme if α−1 OX∗ → OX∗ is an isomorphism.
Example
If X is an regular scheme, j : U → X an open immersion, set
M = {f ∈ OX /f|U ∈ OU∗ }. The inclusion M ⊂ (OX , ×) defines a
Kato log-structure on X . The monoid M = M/OX∗ of effective
Cartier divisors with support in X \U contains relevant
information for the definition of parabolic bundles. Indeed if
x : spec Ω → X is a geometric point, then M x ' NC(x) , where
C(x) is the set of irreducible components of X \U at x of
codimension 1 in X .
Niels Borne
Parabolic vector bundles
Parabolic bundles on curves
Parabolic bundles in higher dimension
Stacks of roots
Open problems
Parabolic bundles as orbifold bundles
Log geometry
Deligne-Faltings log-structures
Starting from a Kato log-structure (X , α : M → OX ), one gets a
commutative diagram of monoids:
M
α
M/Gm
M
L
/ OX
A1
/ DivX
[A1 /Gm ]
where DivX is the symmetric monoidal fibered category over
f
Xet defined by: DivX (X 0 →
− X ) is the category of pairs (L0 , s0 ),
L0 an invertible sheaf, s0 ∈ Γ(X 0 , L0 ).
Given r ∈ N∗ , one defines, following Olsson, a refined stack of
roots X 1 M/M classifying extensions of the given
r
Deligne-Faltings log-structure along 1r M/M.
Niels Borne
Parabolic vector bundles
Parabolic bundles on curves
Parabolic bundles in higher dimension
Stacks of roots
Open problems
Parabolic bundles as orbifold bundles
Log geometry
Parabolic sheaves on log-schemes
To the given Deligne-Faltings log-structure L : M → DivX , and
r ∈ N∗ , one can associate parabolic sheaves.
For any monoid M, denote by M wt the partially ordered set
whose elements are those of M gp ordered by a ≤ b if there
exists c in M such that a + c = b.
The data of L : M → DivX turns out to be equivalent to the data
wt
of a symmetric monoidal functor Lwt : M → PicX .
A parabolic bundle (with weights multiples of 1r ) is a module
1 wt
morphism relative to Lwt : E· : M → VectX
r
Theorem (B.-Vistoli 2009)
The correspondence between vector bundles on X 1 M/M and
r
parabolic bundles on (X , α) with weights multiples of
Niels Borne
Parabolic vector bundles
1
r
holds.
Parabolic bundles on curves
Parabolic bundles in higher dimension
Stacks of roots
Open problems
Parabolic bundles and connections
Wild parabolic bundles
Parabolic bundle with a logarithmic connection
Let X be a smooth scheme over k , algebraically closed field of
characteristic 0, let D be a simple normal crossings divisor,
U = X \D its complement, and x be a geometric point of U.
Esnault and Hai have defined (in greater generality) the
category FC(U) of finite integrable connections on U.
Problem
Give an explicit equivalence between FC(U) and F Par(X , D).
In dimension 1, various authors have considered a notion of
parabolic bundle on (X , D) endowed with a logarithmic
connection: the compatibility condition is that the weights of the
parabolic structure are the eigenvalues of the residues. These
correspond to the (holomorphic) connections on the stacks of
roots, and could be used as a bridge between FC(U) and
F Par(X , D).
Niels Borne
Parabolic vector bundles
Parabolic bundles on curves
Parabolic bundles in higher dimension
Stacks of roots
Open problems
Parabolic bundles and connections
Wild parabolic bundles
Nori’s definition
Definition
Let X be a smooth projective curve, D a reduced effective
[
divisor. For x in the support of D, and Kx = Frac(O
X ,x ), let Kx
be an algebraic closure, of ring of integers Rx . One sets
Z = spec(⊕x∈|D| Kx ), Y1 = X \D et Y2 = spec(⊕x∈|D| Rx ).
A parabolic sheaf with rational weights on (X , D) is a sheaf on
the diagram:
f1 ll6 Y1
llll
Z RRRR
RR(
f2
Y2
Problem
How does this definition relates to (wildly ramified) covers?
Over an algebraically closed field of characteristic 0, is it
equivalent to Seshadri’s definition?
Niels Borne
Parabolic vector bundles