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 projective
variety 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 a
very 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 groups
Hp (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 abelian
variety X and assume that its Euler characteristic χ(L) is
nonzero. The following assertions hold:
I
I
I
there exists a unique integer i(L) s.t. Hi(L) (X , L) 6= 0;
if A is an ample line bundle, then the polynomial
P(N) := χ(AN ⊗ L) has real roots and i(L) equals the
number of positive roots counted with multiplicity
Hi(L) (X , L) is the unique irreducible weight one
representation of the theta group of L.
Remark: An almost identical statement holds for a certain
class of simple vector bundles on X . (This follows from work
of Mumford, and Mukai. See my [-, Ann. Math. Que., To
appear].)
Example: Let E be an elliptic curve and X = E × E .
Let x ∈ E , let f1 = {x} × E , let f2 = E × {x}, let ∆ denote
the diagonal, and let γ = f1 + f2 − ∆.
The numerical classes of f1 , f2 , and γ span a three dimensional
subspace of N1 (X )R the real Neron-Severi space of X :
H0
H1
ab − c2 = 0
H2
If χ(L) 6= 0 and L has numerical class af1 + bf2 + cγ then
8
2
>
<0 iff ab − c > 0 and a + b > 0
i(L) = 1 iff ab − c2 < 0
>
:
2 iff ab − c2 > 0 and a + b < 0.
Remark: The index of line bundles with non-vanishing Euler
characteristic is constant on the connected components of the
open 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 ∈W w · Λ+
A weight λ ∈ Λ is said to be regular if λ ∈ W · Λ+ .
Since
W · Λ+ = tw ∈W w · 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 the
line bundle on X with total space G ×B C−λ .
Theorem (Borel-Weil-Bott). If λ is regular, then
(
Vw∗ ·λ if i = `(w ) and λ ∈ w −1 · Λ+
i
H (X , Lλ ) =
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 = 1 − 2 , and α2 = 2 − 3
I
I
weight lattice: Λ = spanZ {λ1 , λ2 }, where
1
1
2
2
λ1 = α1 + α2 and λ2 = α1 + α2 .
3
3
3
3
+
dominant integral weights: Λ = spanZ≥0 {λ1 , λ2 }
B-W-B theorem for X = SL3 (C)/B
S3 = W = hσα1 , σα2 i ⊆ GL(ΛR )
H1
ρ = λ1 + λ2
H0
H2
H1
H3
H2
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 on
X 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 the
form:
∪
Hi(L) (X , L) ⊗ Hi(M) (X , M) −
→ Hi(L⊗M) (X , L ⊗ M).
In general such maps can be zero, but their asymptotic
behaviour is more uniform.
Theorem (-, Int. J. Math., 2014). In the setting just
described, there exists an n0 > 0 such that the cup-product
Hi(L) (X , Tx∗ (Ln )) ⊗ Hi(M) (X , M n )
∪
−
→ Hi(L⊗M) (X , Tx∗ (Ln ) ⊗ M n )
is nonzero and surjective for every n ≥ n0 and every x ∈ X .
Equivalently, the vector bundle
Ri(L⊗M)
((p1 + p2 )∗ Ln ⊗ p2∗ M n )
p1 ∗
is globally generated for all n ≥ n0 .
Remarks.
I The relationship between the family of cup-products
∪(Tx∗ L, M), for x ∈ X ,
and the vector bundle
Ri(L⊗M)
((p1 + p2 )∗ L ⊗ p2∗ M)
p1 ∗
I
generalizes work of Pareschi and Pareschi-Popa where
they prove and refine a conjecture of Lazarsfeld related to
the property Np for abelian varieties
More general versions of these results apply to higher
rank 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 the
property that 0 ≤ p + q ≤ g . There exists simple (complex)
abelian varieties, of dimension g , which admit line bundles L
and 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 of
higher rank simple vector bundles with similar cohomological
properties.
Idea of proof
Important in the proof of our main result, concerning the
asymptotic nature of cup-product problems, is:
Theorem (-, Int. J. Math., 2014). Let Y be an abelian
variety. Let L and F denote, respectively, a non-degenerate
line bundle and a coherent sheaf on Y . There exists an n0 > 0
with 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 and
i(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):
n/m
X −−→ X
n∗ L ↔ m∗ L
I
I
I
b (L)
The data (X , L) determines an adelic theta group G
i
∗
which acts on the direct limit lim H (X , n L)
If χ(L) 6= 0, then lim H
→
i(L)
→
∗
(X , n L) is an irreducible
b (L)-module
G
b
The adelic theta groups G(L)
exhibit some functorial
properties 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 generalized
Cartan matrix
I B ⊆ G is a standard Borel subgroup
I X = G/B
Remarks.
I In general X is an infinite dimensional variety and carries
the structure of an ind-variety
I When G is determined by an extended Cartan matrix, X is
related to the moduli space of vector bundles on algebraic
curves with parabolic structure
Regular weights λ of G determine line bundles Lλ on X and
there is a Borel-Weil-Bott type theorem:
If λ ∈ W · Λ+ , then
(
Vw∗ ·λ
i
H (X, Lλ ) =
0
if i = `(w ) and λ ∈ w −1 · Λ+
otherwise.
Q: What kinds of positivity/cohomological results, applicable
for ind-varieties for instance, could be useful in studying
cup-product problems on X?