Analogies amongst vector bundles on G/B and Abelian varietiesabelian varieties, of dimension g,...

Analogies amongst vector bundles on G/B

and Abelian varieties

Nathan Grieve

Cup-product problems on projective varieties

Q: Suppose that L and M are line bundles on a projectivevariety X . What can we say about the cup-product map

Hp(X , L)⊗ Hq(X ,M)∪−→ Hp+q(X , L⊗M)?

For instance is it nonzero?

Special case: Suppose that L and M are positive powers of avery ample line bundle A on X . Cup-product maps of the form:

H0(X ,A⊗m)⊗ H0(X ,A⊗n)∪−→ H0(X ,A⊗m+n)

are related to the syzygies of X .

In general: When p, q > 0 we should place conditions on L,M , and L⊗M to ensure that the cohomology groupsHp(X , L), Hq(X ,M), and Hp+q(X , L⊗M) are non-zero.

Mumford’s index theorem

Theorem (Mumford). Let L be a line bundle on an abelianvariety X and assume that its Euler characteristic χ(L) isnonzero. The following assertions hold:

I there exists a unique integer i(L) s.t. Hi(L)(X , L) 6= 0;

I if A is an ample line bundle, then the polynomialP(N) := χ(AN ⊗ L) has real roots and i(L) equals thenumber of positive roots counted with multiplicity

I Hi(L)(X , L) is the unique irreducible weight onerepresentation of the theta group of L.

Remark: An almost identical statement holds for a certainclass of simple vector bundles on X . (This follows from workof Mumford, and Mukai. See my [-, Ann. Math. Que., Toappear].)

Example: Let E be an elliptic curve and X = E × E .

Let x ∈ E , let f1 = {x} × E , let f2 = E × {x}, let ∆ denotethe diagonal, and let γ = f1 + f2 −∆.

The numerical classes of f1, f2, and γ span a three dimensionalsubspace of N1(X )R the real Neron-Severi space of X :

H 0

H 2

H1

ab − c2 = 0

If χ(L) 6= 0 and L has numerical class af1 + bf2 + cγ then

i(L) =

8

>

<

>

:

0 iff ab − c2 > 0 and a + b > 0

1 iff ab − c2 < 0

2 iff ab − c2 > 0 and a + b < 0.

Remark: The index of line bundles with non-vanishing Eulercharacteristic is constant on the connected components of theopen subset of N1(X )R determined by the non-vanishing of χ.

The Borel-Weil-Bott theorem

Set-up:

I G is a semi-simple complex algebraic group

I T ⊆ G a maximal torus

I B ⊆ G a Borel subgroup containing T

I Λ = Hom(T ,C×) the weight lattice

I W = NG (T )/ZG (T ) the Weyl group

I Λ+ the collection of dominant integral weights

A key point: Under the dot action of W on Λ, Λ+

decomposes as a disjoint union:

W · Λ+ = tw∈Ww · Λ+

A weight λ ∈ Λ is said to be regular if λ ∈ W · Λ+.Since

W · Λ+ = tw∈Ww · Λ+,

if λ is regular then λ ∈ w · Λ+ for some unique w ∈ W .

Let X = G/B , suppose that λ ∈ Λ, and suppose that Lλ is theline bundle on X with total space G ×B C−λ.

Theorem (Borel-Weil-Bott). If λ is regular, then

Hi(X , Lλ) =

{V ∗w ·λ if i = `(w) and λ ∈ w−1 · Λ+

0 otherwise.

If λ is not regular, then Hi(X , Lλ) = 0 for all i .

Example: Consider the case:

X = SL3(C)/B = {(p, `) ∈ P2 × P2∗ : p ∈ `}.I standard simple roots of sl3(C):

α1 = ε1 − ε2, and α2 = ε2 − ε3I weight lattice: Λ = spanZ{λ1, λ2}, where

λ1 =2

3α1 +

1

3α2 and λ2 =

1

3α1 +

2

3α2.

I dominant integral weights: Λ+ = spanZ≥0{λ1, λ2}

H 0

H 1

H 1

H 2H 3

H 2

B-W-B theorem for X = SL3(C)/B

S3 = W = 〈σα1 , σα2 〉 ⊆ GL(ΛR)

ρ = λ1 + λ2

σ · λ = σ(λ + ρ) − ρ

Cup-product problems on abelian varieties

Let X be an abelian variety, and fix line bundles L and M onX satisfying the conditions that:

I χ(L) 6= 0, χ(M) 6= 0, and χ(L⊗M) 6= 0 and

I i(L⊗M) = i(L) + i(M).

The cup-product problems that we want to study have theform:

Hi(L)(X , L)⊗ Hi(M)(X ,M)∪−→ Hi(L⊗M)(X , L⊗M).

In general such maps can be zero, but their asymptoticbehaviour is more uniform.

Theorem (-, Int. J. Math., 2014). In the setting justdescribed, there exists an n0 > 0 such that the cup-product

Hi(L)(X ,T ∗x (Ln))⊗ Hi(M)(X ,Mn)∪−→ Hi(L⊗M)(X ,T ∗x (Ln)⊗Mn)

is nonzero and surjective for every n ≥ n0 and every x ∈ X .Equivalently, the vector bundle

Ri(L⊗M)p1∗ ((p1 + p2)∗Ln ⊗ p∗2M

n)

is globally generated for all n ≥ n0.

Remarks.

I The relationship between the family of cup-products

∪(T ∗x L,M), for x ∈ X ,

and the vector bundle

Ri(L⊗M)p1∗ ((p1 + p2)∗L⊗ p∗2M)

generalizes work of Pareschi and Pareschi-Popa wherethey prove and refine a conjecture of Lazarsfeld related tothe property Np for abelian varieties

I More general versions of these results apply to higherrank vector bundles

Existence of nontrivial problems

Using a construction of Shimura and Albert we prove:

Theorem (-, Int. J. Math., 2014). Let p, q ∈ Z≥0 with theproperty that 0 ≤ p + q ≤ g . There exists simple (complex)abelian varieties, of dimension g , which admit line bundles Land M such that:

I χ(L), χ(M), and χ(L⊗M) are nonzero, and

I i(L) = p, i(M) = q, and i(L⊗M) = p + q.

Remark. A more general version works to provide examples ofhigher rank simple vector bundles with similar cohomologicalproperties.

Idea of proof

Important in the proof of our main result, concerning theasymptotic nature of cup-product problems, is:

Theorem (-, Int. J. Math., 2014). Let Y be an abelianvariety. Let L and F denote, respectively, a non-degenerateline bundle and a coherent sheaf on Y . There exists an n0 > 0with the property that, for all isogenies f : X → Y , we have

Hj(X , f ∗(F ⊗ Ln)⊗ α) = 0

for all j > i(L), for all n ≥ n0, and for all α ∈ Pic0(X ).

Corollary (-, Int. J. Math., 2014). If χ(L) 6= 0 andi(L) ≤ q then L is naively q-ample.

Infinite dimensional analogues

Setting of abelian varieties

Fix an abelian variety X and a line bundle L on X .

Consider the polarized tower associated to the pair (X , L):

Xn/m−−→ X

n∗L↔ m∗L

I The data (X , L) determines an adelic theta group G (L)which acts on the direct limit lim

→Hi(X , n∗L)

I If χ(L) 6= 0, then lim→

Hi(L)(X , n∗L) is an irreducible

G (L)-module

I The adelic theta groups G(L) exhibit some functorialproperties which are not shared by the theta groups G(L).(See my [-, Ann. Math. Que., To appear].)

Setting of infinite dimensional Flag varieties

Set-up:

I G is a Kac-Moody group, determined by a generalizedCartan matrix

I B ⊆ G is a standard Borel subgroup

I X = G/BRemarks.

I In general X is an infinite dimensional variety and carriesthe structure of an ind-variety

I When G is determined by an extended Cartan matrix, X isrelated to the moduli space of vector bundles on algebraiccurves with parabolic structure

Regular weights λ of G determine line bundles Lλ on X andthere is a Borel-Weil-Bott type theorem:

If λ ∈ W · Λ+, then

Hi(X, Lλ) =

{V ∗w ·λ if i = `(w) and λ ∈ w−1 · Λ+

0 otherwise.

Q: What kinds of positivity/cohomological results, applicablefor ind-varieties for instance, could be useful in studyingcup-product problems on X?