Let be an abelian variety over the algebraically closed field
. In the previous post, we studied the Picard scheme
, or rather its connected component
at the identity. The main result was that
was itself an abelian variety (in particular, smooth) of the same dimension as
, which parametrizes precisely the translation-invariant line bundles on
.
We also saw how to construct isogenies between and
. Given an ample line bundle
on
, the map
is an isogeny. Such maps were in fact the basic tool in proving the above result.
The goal of this post is to show that the contravariant functor
from abelian varieties over to abelian varieties over
, is a well-behaved duality theory. In particular, any abelian variety is canonically isomorphic to its bidual. (This explains why the double Picard functor on a general variety is the universal abelian variety generated by that variety, the so-called Albanese variety.) In fact, we won’t quite finish the proof in this post, but we will finish the most important step: the computation of the cohomology of the universal line bundle on
.
Motivated by this, we set the notation:
Definition 11 We write
for
.
The main reference for this post is Mumford’s Abelian varieties.
10. The Poincaré line bundle and the biduality map
The first step in understanding the biduality of abelian varieties is to understand the universal line bundle on . By definition,
parametrizes line bundles on
algebraically equivalent to zero, so there is a universal line bundle
, called the Poincaré line bundle, on
.
The line bundle has the property that
is trivialized on
and
, and as
varies, the various restrictions of
to
range exactly over the line bundles on
algebraically equivalent to zero: this is precisely the definition of
.
As we’ve observed before, there is quite a bit of symmetry in this. Instead of regarding as a family of line bundles on
, parametrized by
, we can regard it as a family of line bundles on the dual
parametrized by
. The result is that we get a canonical biduality map
classifying the Poincaré bundle. This is a pointed map, hence a morphism of abelian varieties. The main goal of this post and the next is to show:
Theorem 12 For any abelian variety
, the biduality map
is an isomorphism.
In particular, this implies a non-obvious fact about the Poincaré bundle : as one restricts to the various fibers
(for
), one gets every translation-invariant line bundle on
exactly once.
The strategy in proving the above theorem is to show first that the biduality map is finite, by a diagram chase and the isogenies constructed before. Next, we’ll show that the Poincaré bundle isn’t “redundant” on either factor by showing that its Euler characteristic is
: this will imply that the biduality map is actually an isomorphism.
11. Finiteness of the biduality map
The first goal is to show that the biduality map is a finite morphism: that is, it cannot annihilate a nontrivial abelian subvariety
. In other words, in terms of the Poincaré bundle, it states that for any
, the restricted bundle
is still nontrivial.
To see this, observe that is a family of line bundles on
: it’s classified by the map
dual to the inclusion . The claim is that not only is this family nontrivial, but also that it hits every translation-invariant line bundle on
. That is, every translation-invariant line bundle on
extends to
. In other words:
Proposition 13 If
is a map of abelian varieties with finite kernel, then
is surjective.
Proof: Given an ample line bundle , we have a morphism
which classifies the bundle , for
the multiplication.
The restriction to
is still ample, since
is finite, and we have a commutative diagram:
Since is surjective, it follows that
is surjective.
This in particular shows that the biduality map must be finite.
12. The cohomology of
Our next goal is to compute the cohomology of the Poincaré bundle on
. (In particular,
itself is not algebraically equivalent to zero!) We can do this using the Leray spectral sequence and the categorified “orthogonality of characters,” although the argument is a bit technical, and we’ll split it into two sections.
Namely, is a family of line bundles on
, parametrized by
. By “orthogonality of characters,” each fiber
has vanishing cohomology except when
. It follows that the complex
which lives in the derived category of sheaves on , is concentrated at the identity
.
If is the local ring at
, we can localize this complex to get a complex
of
-modules. The general yoga of base-change tells us that
is a perfect complex, and the (derived) tensor product
is precisely the cohomology of
along the fiber: that is,
.
Let . Since
is concentrated in dimensions
, we conclude by Nakayama’s lemma that the cohomology of
itself is concentrated in dimensions
. Moreover, the cohomology of
consists of artinian
-modules, because it is supported at the origin.
That already buys us something. Roughly, there can’t be any cohomology close to zero, because then the derived tensor product with would blow that up into negative dimensions. (The grading is cohomological here.) In fact, we have a precise statement:
Lemma 14 If
is a regular local ring of dimension
, and
is a perfect complex of
-modules with artinian cohomology such that the (derived) tensor product
is cohomologically concentrated in dimensions
, then
has cohomology concentrated in dimensions
.
Proof: Induction on . When
and
is a field, it is evident. Assume that it is true for regular local rings of dimension
.
We just have to prove that there is no cohomology below , by Nakayama’s lemma. Given a regular parameter
(in the maximal ideal), we can form
, which is a perfect complex of
-modules satisfying the same hypotheses. In particular,
has no cohomology below dimension
by the inductive hypothesis. Now consider the cofiber sequence
and the exact sequence in cohomology
For , this means that multiplication by
is injective on
; since these are artinian modules, they must vanish.
13. The cohomology of : part 2
Keep the notation of the previous section. Our conclusion is that the derived push-forward must be supported at the point
, and concentrated in cohomological dimension
. The claim is that the cohomology in dimension
is precisely the ground field
.
In fact, we start by noting:
by Serre duality (recall that ). It follows that
has the property that
is generated by one element. Moreover,
To see that is in fact
, we will use a bit of local duality theory, as explained in this post. Namely, the statement is that the category of artinian modules over
is dual to itself, via the local duality functor
In fact, is given by the derived maps from
to
, up to a cohomological shift.
In our case, the (artinian) module has the property that
is generated by one element. That doesn’t mean that
, though (not even that along with
: take
over
). We need to use an additional feature that can be seen using the local duality. The strategy is going to be to show that
is isomorphic to
, because
and because the surjection
can’t be lifted further.
Let be a finite complex of projectives quasi-isomorphic to
(i.e. a “cofibrant replacement”). We observe that for any artinian
-algebra
, we have
, which cannot surject onto
simply because
cannot be trivialized beyond
.
That last statement is a consequence of the fact that is the solution to a moduli problem: if
pulled
back to a trivial bundle, then by definition it would have to be constant at the origin.
Now, we can write
for the local duality functor from artinian
-modules to itself. In fact,
The conclusion on is that:
maps to
nontrivially. In fact, taking
, we find that
is one-dimensional, so
.
- In particular,
for
an ideal (containing some power of
.
- However, the map
cannot be lifted under the map
for
any local artinian
-algebra. This proves
and
.
- Dualizing, we find that
.
The conclusion that we get (from this calculation plus the degenerate Leray spectral sequence) is:
Theorem 15 The cohomology of the Poincaré bundle is given by:
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