The Fourier-Mukai transform made easy

TL;DR

This paper introduces a simplified approach to the Fourier-Mukai transform on abelian varieties, facilitating easier understanding and proofs of key theorems related to sheaf properties and global generation.

Contribution

It presents a modified definition of the Fourier-Mukai transform that simplifies formula derivation and provides concise proofs for important theorems in algebraic geometry.

Findings

Simplified Fourier-Mukai transform formula

Short proof of GV-sheaves characterization

Proof that M-regularity implies global generation

Abstract

We propose a slightly modified definition for the Fourier-Mukai transform (on abelian varieties) that makes it much easier to remember various formulas. As an application, we give relatively short proofs for two important theorems: the characterization of GV-sheaves in terms of vanishing, due to Hacon; and fact that M-regularity implies (continuous) global generation, due to Pareschi and Popa.

Equations

$$\mathbf{R}{\Phi }_{P_A}:D_{coh}^b({\mathcal{O}}_A)\to D_{coh}^b({\mathcal{O}}_{A^{\vee }})$$

$$\mathbf{R}\Delta_{X}=\mathbf{R}\mathcal{H}\mathit{om}\bigl{(}-,\omega_{X/k}[\dim X]\bigr{)}\colon D_{\mathit{coh}}^{\mathit{b}}(\mathscr{O}_{X})\to D_{\mathit{coh}}^{\mathit{b}}(\mathscr{O}_{X})^{\mathit{op}}$$

$${\mathrm{FM}}_A=\mathbf{R}{\Phi }_{P_A}\circ \mathbf{R}{\Delta }_A:D_{coh}^b({\mathcal{O}}_A)\to D_{coh}^b({\mathcal{O}}_{A^{\vee }}{)}^{op}$$

$$\operatorname{\mathsf{FM}}_{A}\bigl{(}k(0)\bigr{)}=\mathscr{O}_{A^{\vee}}\quad\text{and}\quad\operatorname{\mathsf{FM}}_{A}(\mathscr{O}_{A})=k(0).$$

$${\mathrm{FM}}_A:D_{coh}^b({\mathcal{O}}_A)\to D_{coh}^b({\mathcal{O}}_{A^{\vee }}{)}^{op}$$

$${\mathrm{FM}}_B\circ \mathbf{R}{f}_{\ast }=\mathbf{L}{\hat{f}}^{\ast }\circ {\mathrm{FM}}_A\text{and}{\mathrm{FM}}_A\circ \mathbf{L}{f}^{\ast }=\mathbf{R}{\hat{f}}_{\ast }\circ {\mathrm{FM}}_B,$$

$${\mathrm{FM}}_A\circ (t_a{)}_{\ast }=(P_a\otimes -)\circ {\mathrm{FM}}_A\text{and}{\mathrm{FM}}_A\circ (P_{\alpha }\otimes -)=(t_{\alpha }{)}_{\ast }\circ {\mathrm{FM}}_A,$$

$$\operatorname{\mathsf{FM}}_{A}\bigl{(}k(a)\bigr{)}=P_{a}\quad\text{and}\quad\operatorname{\mathsf{FM}}_{A}(P_{\alpha})=k(\alpha),$$

$${\mathrm{FM}}_A(\mathcal{F})=\hat{\mathcal{F}}$$

$$\bigl{(}\operatorname{Supp}\hat{\mathscr{F}}\bigr{)}(k)=\bigl{\{}\thinspace\alpha\in A^{\vee}(k)\thinspace\thinspace\big{|}\thinspace\thinspace H^{0}\bigl{(}A,\mathscr{F}\otimes P_{\alpha}^{-1}\bigr{)}\neq 0\thinspace\bigr{\}}.$$

$$S^{j}(A,\mathscr{F})(k)=\bigl{\{}\thinspace\alpha\in A^{\vee}(k)\thinspace\thinspace\big{|}\thinspace\thinspace H^{j}(A,\mathscr{F}\otimes P_{\alpha}^{-1})\neq 0\thinspace\bigr{\}}.$$

$$S^0(A,\mathcal{F})\supseteq S^1(A,\mathcal{F})\supseteq \cdots \supseteq S^{\dim A}(A,\mathcal{F}),$$

$${\varphi }_L:A\to A^{\vee },$$

$${\varphi }_L^{\ast }\hat{L}=i^{\ast }L\otimes H^0(A,L{)}^{\ast },$$

$$\chi (A,\mathcal{F})=\sum _{j=0}^{\dim A}(-1{)}^j\dim H^j(A,\mathcal{F}).$$

$${\mathrm{FM}}_B(f_{\ast }\mathcal{F})={\hat{f}}^{\ast }\hat{\mathcal{F}},$$

$$H^i(A^{\prime },L^{\prime }\otimes {\phi }^{\ast }\mathcal{F})=0\text{for all i>0.}$$

$$\bigoplus_{\alpha\in U(k)}H^{0}\bigl{(}A,\mathscr{F}\otimes P_{\alpha}^{-1}\bigr{)}\otimes P_{\alpha}\to\mathscr{F}$$

$$m:A\times A\to A,i:A\to A,e:\mathrm{Spec}k\to A$$

$$P_{(a,\alpha)}=P_{A}\big{|}_{(a,\alpha)}$$

$$P_{(a,\alpha)}=P_{a}\big{|}_{\alpha}=P_{\alpha}\big{|}_{a}=P_{(-a,-\alpha)}=P_{-\alpha}\big{|}_{-a}=P_{-a}\big{|}_{-\alpha}.$$

$$\mathbf{R}{\Phi }_{P_A}:D_{coh}^b({\mathcal{O}}_A)\to D_{coh}^b({\mathcal{O}}_{A^{\vee }}),$$

$$A\times A^{\vee }$$

$$\mathbf{R}\Delta_{A}\bigl{(}k(0)\bigr{)}=e_{\ast}\mathbf{R}\Delta_{\operatorname{Spec}k}\bigl{(}\mathscr{O}_{\operatorname{Spec}k}\bigr{)}=e_{\ast}\mathscr{O}_{\operatorname{Spec}k}=k(0),$$

$$\operatorname{\mathsf{FM}}_{A}\bigl{(}k(0)\bigr{)}=\mathbf{R}\Phi_{P_{A}}\bigl{(}k(0)\bigr{)}=\mathscr{O}_{A}.$$

$$\operatorname{\mathsf{FM}}_{A}(\mathscr{O}_{A})=\mathbf{R}\Phi_{P_{A}}\Bigl{(}\omega_{A}[\dim A]\Bigr{)}=\mathbf{R}(p_{2})_{\ast}\Bigl{(}P_{A}\otimes p^{\ast}_{1}\omega_{A}[\dim A]\Bigr{)},$$

$$A$$

$$H^{n}(A,P_{\alpha})\to H^{n}\bigl{(}A\times A,m^{\ast}P_{\alpha}\bigr{)}\to H^{n}(A,P_{\alpha}).$$

$$H^{n}\bigl{(}A\times A,m^{\ast}P_{\alpha}\bigr{)}=\bigoplus_{i+j=n}H^{i}(A,P_{\alpha})\otimes H^{j}(A,P_{\alpha}),$$

$$\mathscr{F}_{j}=R^{j}(p_{2})_{\ast}\Bigl{(}P_{A}\otimes p^{\ast}_{1}\omega_{A}[\dim A]\Bigr{)}=R^{\dim A+j}(p_{2})_{\ast}\bigl{(}P_{A}\otimes p^{\ast}_{1}\omega_{A}\bigr{)}$$

Full text

A. Introduction

1.

The Fourier-Mukai transform, first introduced by Mukai [Mukai], is an equivalence of $k$-linear triangulated categories

[TABLE]

between the bounded derived category of coherent sheaves on the abelian variety $A$, and that on the dual abelian variety $A^{\vee}=\operatorname{Pic}_{A/k}^{0}$. It is defined by pulling back to $A\times A^{\vee}$, tensoring by the Poincaré bundle $P_{A}$, and then pushing forward to $A^{\vee}$.

2.

The purpose of this note is to propose a slightly different definition for the Fourier-Mukai transform that makes it easier to remember various formulas. When $X$ is an equidimensional and nonsingular algebraic variety over $k$, we denote by

[TABLE]

the (contravariant) Grothendieck duality functor. In the case $X=\operatorname{Spec}k$, we shall use the simplified notation $\mathbf{R}\Delta_{k}$.

Definition 2.1**.**

The exact functor

[TABLE]

is called the symmetric Fourier-Mukai transform.

The word “symmetric” will be justified by the results below. Note that $\operatorname{\mathsf{FM}}_{A}$ is a contravariant functor; this turns out to be quite useful in practice, for instance in the study of GV-sheaves (see §6).

3.

It is clear from Mukai’s result [Mukai, Thm 2.2] that $\operatorname{\mathsf{FM}}_{A}$ is also an equivalence of categories. One advantage of the new definition is that it respects the symmetry between the two abelian varieties $A$ and $A^{\vee}$. For example, one can show that

[TABLE]

Here $k(0)=e_{\ast}\mathscr{O}_{\operatorname{Spec}k}$ means the structure sheaf of the closed point $0\in A(k)$; we use the same notation also on $A^{\vee}$. The following theorem further justifies the name “symmetric Fourier-Mukai transform”.

Theorem 3.2**.**

The composed functors $\operatorname{\mathsf{FM}}_{A^{\vee}}\circ\operatorname{\mathsf{FM}}_{A}$ and $\operatorname{\mathsf{FM}}_{A}\circ\operatorname{\mathsf{FM}}_{A^{\vee}}$ are naturally isomorphic to the identity. In other words,

[TABLE]

is an equivalence of categories, with quasi-inverse $\operatorname{\mathsf{FM}}_{A^{\vee}}$.

This looks much simpler than Mukai’s version [Mukai, Thm. 2.2].

4.

The effect of pulling back or pushing forward by a homomorphism between abelian varieties is easy to describe; for the case of isogenies, see [Mukai, (3.4)], and for the general case, [CJ, Proposition 2.3].

Proposition 4.1**.**

Let $f\colon A\to B$ be a homomorphism of abelian varieties over $k$. Then one has natural isomorphisms of functors

[TABLE]

where $\hat{f}\colon B^{\vee}\to A^{\vee}$ is the induced homomorphism between the dual abelian varieties.

5.

The symmetric Fourier-Mukai transform exchanges translations and tensoring by the corresponding line bundles. Any closed point $a\in A(k)$ determines a translation morphism $t_{a}\colon A\to A$; on closed points, it is given by the formula $t_{a}(x)=a+x$. Compare the following result with [Mukai, (3.1)].

Proposition 5.1**.**

Let $a\in A(k)$ and $\alpha\in A^{\vee}(k)$ be closed points. Then one has natural isomorphisms of functors

[TABLE]

where $P_{a}$ on $A$, and $P_{\alpha}$ on $A^{\vee}$, are the corresponding line bundles.

Together with (3.1), this leads to the pleasant formulas

[TABLE]

for any pair of closed points $a\in A(k)$ and $\alpha\in A^{\vee}(k)$.

6.

The symmetric Fourier-Mukai transform has the following simple effect on the standard t-structure on $D_{\mathit{coh}}^{\mathit{b}}(\mathscr{O}_{A})$.

Proposition 6.1**.**

If $K\in D_{\mathit{coh}}^{\geq n}(\mathscr{O}_{A})$, then $\operatorname{\mathsf{FM}}_{A}(K)\in D_{\mathit{coh}}^{\leq-n}(\mathscr{O}_{A^{\vee}})$.

In particular, the symmetric Fourier-Mukai transform of a coherent sheaf is a complex of coherent sheaves concentrated in non-positive degrees. This motivates the following definition, first made by Pareschi and Popa [PP, Definition 3.1].

Definition 6.2**.**

A coherent sheaf $\mathscr{F}\in\operatorname{Coh}(\mathscr{O}_{A})$ is called a GV-sheaf if

[TABLE]

for a coherent sheaf $\hat{\mathscr{F}}\in\operatorname{Coh}(\mathscr{O}_{A^{\vee}})$. If $\hat{\mathscr{F}}$ is torsion-free, then $\mathscr{F}$ is called M-regular.

Our definition of M-regularity differs from the original one in [PP-regI, Def. 2.1]; the two definitions are equivalent by [PP-regIII, Thm. 2.8].

7.

Theorem 3.2 shows that $\operatorname{\mathsf{FM}}_{A^{\vee}}(\hat{\mathscr{F}})=\mathscr{F}$, and so the coherent sheaf $\hat{\mathscr{F}}$ is again a GV-sheaf on $A^{\vee}$. The property of being a GV-sheaf is therefore self-dual. Using Proposition 4.1 and Proposition 5.1, one shows that the set of closed points in the support of the coherent sheaf $\hat{\mathscr{F}}$ is equal to

[TABLE]

The minus sign is consistent with the formulas in (5.2).

8.

GV-sheaves have several other surprising properties. Let $\mathscr{F}$ be a coherent sheaf on $A$. For each $j\geq 0$, one has a reduced closed subscheme $S^{j}(A,\mathscr{F})\subseteq A^{\vee}$; its set of closed points is given by

[TABLE]

If $\mathscr{F}$ is a GV-sheaf, then $S^{0}(A,\mathscr{F})=\operatorname{Supp}\hat{\mathscr{F}}$, according to (7.1). The following result is originally due to Hacon [Hacon, Cor. 3.2].

Proposition 8.1**.**

If $\mathscr{F}$ is a GV-sheaf on $A$, then

[TABLE]

and moreover, one has $\operatorname{codim}S^{j}(A,\mathscr{F})\geq j$ for all $j\geq 0$.

9.

Another advantage of our definition is that it preserves positivity. Recall that an ample line bundle $L$ on the abelian variety $A$ gives rise to a homomorphism

[TABLE]

with the property that $t_{a}^{\ast}L=L\otimes P_{\phi_{L}(a)}$ for every closed point $a\in A(k)$. The following result is a version of [Mukai, Prop. 3.1].

Proposition 9.2**.**

Let $L$ be an ample line bundle on $A$. Then $\hat{L}$ is an ample vector bundle of rank $\dim H^{0}(A,L)$, and one has

[TABLE]

where $i\colon A\to A$ is the inversion morphism.

Note that $\operatorname{rk}\hat{L}=\dim H^{0}(A,L)$ whereas $\operatorname{rk}L=\dim H^{0}(A^{\vee},\hat{L})$; at least for ample line bundles, the symmetric Fourier-Mukai transform therefore interchanges “rank” and “dimension of the space of global sections”. The generalization to arbitrary GV-sheaves is that the generic rank of $\hat{\mathscr{F}}$ is equal to the Euler characteristic

[TABLE]

This follows from Proposition 8.1; compare also [Mukai, Cor. 2.8].

10.

Finally, we note the following useful property of GV-sheaves, which follows almost immediately from Proposition 6.1. As far as I know, this is a new result.

Proposition 10.1**.**

Let $f\colon A\to B$ be a homomorphism of abelian varieties, and let $\mathscr{F}$ be a GV-sheaf on $A$. If $f_{\ast}\mathscr{F}$ is again a GV-sheaf on $B$, then

[TABLE]

where $\hat{f}\colon B^{\vee}\to A^{\vee}$ is the induced homomorphism between the dual abelian varieties.

11.

Although one can deduce all the results above from what is in Mukai’s paper, we are going to prove everything from scratch. This will hopefully convince the reader that the new definition is also easy to work with in practice. As an application, we present relatively short proofs for two important results about GV-sheaves and M-regular sheaves. The following theorem by Hacon [Hacon, Thm 1.2] relates the GV-property to vanishing theorems for ample line bundles.

Theorem 11.1**.**

A coherent sheaf $\mathscr{F}\in\operatorname{Coh}(\mathscr{O}_{A})$ is a GV-sheaf if, and only if, for every isogeny $\varphi\colon A^{\prime}\to A$ and every ample line bundle $L^{\prime}$ on $A^{\prime}$, one has

[TABLE]

12.

Recall that a coherent sheaf $\mathscr{F}$ on an abelian variety is said to be continuously globally generated [PP-regI, Def 2.10] if for every nonempty Zariski-open subset $U\subseteq A^{\vee}$, the evaluation morphism

[TABLE]

is surjective. The first half of the following result is due to Pareschi and Popa [PP-regI, Prop 2.13], the second half to Debarre [Debarre, Prop 3.1].

Theorem 12.1**.**

Let $\mathscr{F}\in\operatorname{Coh}(\mathscr{O}_{A})$ be an M-regular coherent sheaf on $A$.

1. (a)

$\mathscr{F}$* is continuously globally generated.* 2. (b)

There is an isogeny $\varphi\colon A^{\prime}\to A$ such that $\varphi^{\ast}\mathscr{F}$ is globally generated.

In fact, one can prove more generally that if $\mathscr{F}$ is a torsion-free coherent sheaf on $A$, then $\mathcal{H}^{0}\operatorname{\mathsf{FM}}_{A}(\mathscr{F})$ is continuously globally generated, and globally generated after a suitable isogeny (see Theorem 35.1 below). This is again a new result.

13.

I thank Luigi Lombardi and Mihnea Popa for listening to my ideas about the symmetric Fourier-Mukai transform, and for their comments about the first draft of this note. While working out the details of the construction, I was partially supported by grant DMS-1404947 from the National Science Foundation, and by a Centennial Fellowship from the American Mathematical Society.

B. Notation

14.

Let $A$ be an abelian variety over an algebraically closed field $k$. We denote by

[TABLE]

the $k$-morphisms representing composition, inversion, and the identity of the group scheme $A$; we write $0\in A(k)$ for the closed point corresponding to $e$.

We use the notation $A^{\vee}=\operatorname{Pic}_{A/k}^{0}$ for the dual abelian variety. On the product $A\times A^{\vee}$, we have the Poincaré bundle $P_{A}$, normalized by the condition that its pullback by $e\times\operatorname{id}$ is trivial; the Poincaré correspondence consists of the line bundle $P_{A}$, together with fixed trivializations of $(e\times\operatorname{id})^{\ast}P_{A}$ and $(\operatorname{id}\times e)^{\ast}P_{A}$.

Note that $\operatorname{Pic}_{A^{\vee}/k}^{0}$ is canonically isomorphic to $A$, and that the Poincaré correspondence on $A^{\vee}\times A$ is the pullback of the Poincaré correspondence on $A\times A^{\vee}$, under the automorphism that swaps the two factors of the product.

15.

For a closed point $\alpha\in A^{\vee}(k)$, we denote by $P_{\alpha}$ the corresponding line bundle on $A$; to be precise, $P_{\alpha}$ is the restriction of $P_{A}$ to the closed subscheme $A\times\{\alpha\}$. Similary, $P_{a}$ means the line bundle on $A^{\vee}$ corresponding to a closed point $a\in A(k)$. We also denote by

[TABLE]

the fiber of the Poincaré bundle at the closed point $(a,\alpha)\in(A\times A^{\vee})(k)$. Using the canonical isomorphism $(i\times i)^{\ast}P_{A}=P_{A}$, we obtain

[TABLE]

All these isomorphisms are uniquely determined by the Poincaré correspondence.

C. Proofs

16.

We use the notation $D_{\mathit{coh}}^{\mathit{b}}(\mathscr{O}_{A})$ for the derived category of cohomologically bounded and coherent complexes of sheaves of $\mathscr{O}_{A}$-modules; this category is equivalent to the (much smaller) bounded derived category of $\operatorname{Coh}(\mathscr{O}_{A})$. In Mukai’s original definition, the Fourier-Mukai transform is the exact functor

[TABLE]

given by the (slightly abbreviated) formula \mathbf{R}\Phi_{P_{A}}=\mathbf{R}(p_{2})_{\ast}\bigl{(}P_{A}\otimes p^{\ast}_{1}\bigr{)}. Here $p_{1}$ and $p_{2}$ are the projections from $A\times A^{\vee}$ to the two factors:

[TABLE]

17.

Let us start by checking the two examples in (3.1). The first one is very easy: Grothendieck duality, applied to the morphism $e\colon\operatorname{Spec}k\to A$, gives

[TABLE]

and so the symmetric Fourier-Mukai transform of $k(0)$ is

[TABLE]

The second isomorphism comes from the fixed trivialization of $(e\times\operatorname{id})^{\ast}P_{A}$.

18.

The other example in (3.1) needs more work. To help us describe

[TABLE]

we first recall the following result about the cohomology of line bundles on abelian varieties [Mumford, §8(vii)].

Lemma 18.2**.**

If $\alpha\in A^{\vee}(k)$ is nontrivial, then $H^{n}(A,P_{\alpha})=0$ for all $n\in\mathbb{N}$.

Proof.

The proof is by induction on $n\geq 0$. The case $n=0$ is easy: if $s$ is a nontrivial global section of $P_{\alpha}$, then $i^{\ast}s$ is a nontrivial global section of $i^{\ast}P_{\alpha}=P_{-\alpha}$, and so $s\otimes i^{\ast}s$ is a nontrivial global section of $P_{\alpha}\otimes P_{-\alpha}=\mathscr{O}_{A}$; this clearly implies $\alpha=0$. To deal with the general case, we factor $\operatorname{id}\colon A\to A$ as

[TABLE]

On the level of cohomology, we obtain a factorization of the identity as

[TABLE]

Because $m^{\ast}P_{\alpha}=p^{\ast}_{1}P_{\alpha}\otimes p^{\ast}_{2}P_{\alpha}$, the Künneth formula gives

[TABLE]

which is zero by induction. The conclusion is that $H^{n}(A,P_{\alpha})=0$. ∎

19.

The remainder of the argument is taken from [Mumford, §13], with a few minor tweaks. Consider the cohomology sheaves

[TABLE]

of the complex in (18.1); they are coherent sheaves on the dual abelian variety $A^{\vee}$. Since the line bundle $P_{A}\otimes p^{\ast}_{1}\omega_{A}$ is flat over $A^{\vee}$, we can apply the base change theorem and Lemma 18.2 to conclude that each $\mathscr{F}_{j}$ is supported on the closed point $0\in A^{\vee}(k)$; being coherent, $\mathscr{F}_{j}$ therefore has finite length. For dimension reasons, we obviously have $\mathscr{F}_{j}=0$ for $j>0$.

20.

The next task is to show that $\mathscr{F}_{j}=0$ also for $j<0$. This can be done by using Serre duality. Consider the Leray spectral sequence

[TABLE]

We have $E_{2}^{i,j}=0$ for $i>0$ (because $\operatorname{Supp}\mathscr{F}_{j}$ is zero-dimensional); the spectral sequence therefore degenerates at $E_{2}$ and gives us isomorphisms

[TABLE]

In particular, this group vanishes for $j>0$. By Serre duality, the $k$-vector space on the left is dual to

[TABLE]

and this vanishes for $j<0$, for the same reason as before. But then H^{0}\bigl{(}A^{\vee},\mathscr{F}_{j}\bigr{)}=0, and because the sheaf $\mathscr{F}_{j}$ has finite length, we see that $\mathscr{F}_{j}=0$ for every $j<0$.

21.

So far, we have shown that the natural morphism

[TABLE]

is an isomorphism. To conclude the proof of (3.1), we need to argue that $\mathscr{F}_{0}=k(0)$. Another application of the base change theorem yields

[TABLE]

By Nakayama’s lemma, it follows that $\mathscr{F}_{0}=\mathscr{O}_{Z}$ for a (maybe nonreduced) closed subscheme $Z\subseteq A^{\vee}$ supported on the closed point $0\in A^{\vee}(k)$. We now use the universal property of the Poincaré correspondence to show that $Z$ is reduced, and hence that $\mathscr{F}_{0}=k(0)$. Let $\mathscr{F}\in\operatorname{Coh}(A^{\vee})$ be an arbitrary coherent sheaf. Grothendieck duality, applied to the second projection $p_{2}\colon A\times A^{\vee}\to A^{\vee}$, gives us an isomorphism

[TABLE]

functorial in $\mathscr{F}$. In particular, the identity endomorphism of $\mathscr{O}_{Z}$ corresponds to a nontrivial morphism $P_{A}\to p^{\ast}_{2}\mathscr{O}_{Z}$, hence to a nontrivial morphism

[TABLE]

Because the restriction of $P_{A}$ to the closed subscheme $A\times\{0\}$ is trivial, Nakayama’s lemma tells us that this morphism is an isomorphism. In other words, the restriction of $P_{A}$ to the subscheme $A\times Z$ is trivial; by the universal property of the Poincaré correspondence, the subscheme in question therefore has to be contained in $A\times\{0\}$, which means exactly that $Z$ is reduced.

22.

Having shown that $\operatorname{\mathsf{FM}}_{A}(\mathscr{O}_{A})=k(0)$, we can now explain why the symmetric Fourier-Mukai transform is an equivalence of categories.

Proof of Theorem 3.2.

Since we can interchange the role of $A$ and $A^{\vee}$, we only need to prove that the functor

[TABLE]

is naturally isomorphic to the identity. To do this, we first consider

[TABLE]

A brief computation using Grothendieck duality shows that this functor is an integral transform, whose kernel is the complex

[TABLE]

on the product $A\times A^{\vee}$. After composing with $\mathbf{R}\Phi_{P_{A^{\vee}}}$ and swapping the order of the factors, we obtain another integral transform, whose kernel is now the complex

[TABLE]

on $A\times A$. Theorem 3.2 will be proved once we show that (22.1) is isomorphic to the structure sheaf of the diagonal in $A\times A$.

Let $s\colon A\times A\to A$ be defined as $s=m\circ(\operatorname{id}\times i)$; the formula on closed points is $s(x,y)=x-y$. Considering $p^{\ast}_{13}P_{A}^{-1}\otimes p^{\ast}_{23}P_{A}$ as a correspondence between $A\times A$ and $A^{\vee}$, and using the universal property of the Poincaré correspondence, we obtain

[TABLE]

We can now apply flat base change in the commutative diagram

[TABLE]

and conclude that the complex in (22.1) is isomorphic to

[TABLE]

where $\Delta\colon A\to A\times A$ is the diagonal embedding. Here we used the fact that the symmetric Fourier-Mukai transform of the structure sheaf $\mathscr{O}_{A^{\vee}}$ is the structure sheaf $k(0)$ of the closed point $0\in A(k)$, by (3.1). ∎

23.

In this section, we investigate how the symmetric Fourier-Mukai transform interacts with pushing forward and pulling back by homomorphisms between abelian varieties. Let $f\colon A\to B$ be a homomorphism, and denote by $\hat{f}\colon B^{\vee}\to A^{\vee}$ the induced homomorphism between the dual abelian varieties.

Proof of Proposition 4.1.

It will be enough to show that

[TABLE]

the second identity in the theorem follows from this with the help of Theorem 3.2. Using the definition of $\operatorname{\mathsf{FM}}_{B}$ and Grothendieck duality, we obtain

[TABLE]

This reduces the problem to proving that

[TABLE]

We make use of the following commutative diagram:

[TABLE]

The universal property of the Poincaré correspondence gives $(f\times\operatorname{id})^{\ast}P_{B}=(\operatorname{id}\times\hat{f})^{\ast}P_{A}$. Using the projection formula and flat base change, we have

[TABLE]

This calculation establishes Proposition 4.1. ∎

24.

In this section, we investigate the effect of translations by closed points.

Proof of Proposition 5.1.

Once again, it suffices to prove that

[TABLE]

because the other identity follows from this with the help of Theorem 3.2. Using Grothendieck duality, we get a natural isomorphism of functors

[TABLE]

and so the problem is reduced to showing that

[TABLE]

We use the following commutative diagram:

[TABLE]

Since $(t_{a}\times\operatorname{id})^{\ast}P_{A}=p^{\ast}_{2}P_{a}\otimes P_{A}$, we have

[TABLE]

which is exactly what we needed. ∎

25.

In this section, we prove the assertions about GV-sheaves in the introduction.

Proof of Proposition 6.1.

The functor $\operatorname{\mathsf{FM}}_{A}$ is exact and contravariant, and so it suffices to show that $\operatorname{\mathsf{FM}}_{A}(\mathscr{F})\in D_{\mathit{coh}}^{\leq 0}(\mathscr{O}_{A^{\vee}})$ for every coherent sheaf $\mathscr{F}\in\operatorname{Coh}(\mathscr{O}_{A})$. Observe that the $j$-th cohomology sheaf of the complex

[TABLE]

is supported in codimension $\geq j+\dim A$; equivalently, the dimension of its support is $\leq-j$. In the spectral sequence

[TABLE]

we therefore have $E_{2}^{i,j}=0$ whenever $i>-j$, for dimension reasons. We conclude that $\mathcal{H}^{n}\operatorname{\mathsf{FM}}_{A}(\mathscr{F})=0$ whenever $n>0$, which is what we needed to show. ∎

Suppose now that $\mathscr{F}$ is a GV-sheaf; as in Definition 6.2, we let $\hat{\mathscr{F}}=\operatorname{\mathsf{FM}}_{A}(\mathscr{F})$. Let us compute the restriction of the coherent sheaf $\hat{\mathscr{F}}$ to a closed point $\alpha\in A^{\vee}(k)$. We can write this restriction in the form

[TABLE]

with $t_{-\alpha}\colon A^{\vee}\to A^{\vee}$ denoting translation by $-\alpha$. We can then combine the identies in Proposition 4.1 and Proposition 5.1 to obtain an isomorphism

[TABLE]

where $p\colon A\to\operatorname{Spec}k$ is the structure morphism. In particular

[TABLE]

and so $\alpha\in A^{\vee}(k)$ belongs to the support of $\hat{\mathscr{F}}$ if and only if $H^{0}(A,\mathscr{F}\otimes P_{-\alpha})\neq 0$, proving (7.1) from the introduction.

26.

This seems like a good place to prove Proposition 8.1 about the loci $S^{j}(A,\mathscr{F})$.

Proof of Proposition 8.1.

Let $\alpha\in A^{\vee}(k)$ be a closed point. It will be convenient to use the symbol $i_{\alpha}=t_{\alpha}\circ e\colon\operatorname{Spec}k\hookrightarrow A^{\vee}$ for the resulting closed embedding. After taking cohomology in (25.1), we obtain

[TABLE]

This implies of course that

[TABLE]

All the assertions in the proposition are now general facts about coherent sheaves on nonsingular varieties. For the convenience of the reader, we recall the necessary result (and its simple proof) in the following paragraph. ∎

27.

Let $X$ be a nonsingular algebraic variety over $k$; for a closed point $x\in X(k)$, we denote by $i_{x}\colon\operatorname{Spec}k\hookrightarrow X$ the resulting closed embedding. For any coherent sheaf $\mathscr{F}$ on $X$, there are closed algebraic subsets $S^{-j}(\mathscr{F})\subseteq X$, indexed by $j\in\mathbb{N}$; on closed points, they are given by the formula

[TABLE]

Here is the result that we used to prove Proposition 8.1.

Lemma 27.1**.**

Let $X$ be a nonsingular algebraic variety over $k$, and let $\mathscr{F}$ be a coherent sheaf on $X$.

1. (a)

One has $S^{-(j+1)}(\mathscr{F})\subseteq S^{-j}(\mathscr{F})$. 2. (b)

Every irreducible component of $S^{-j}(\mathscr{F})$ has codimension $\geq j$.

Proof.

The basic fact is that every finitely generated module over a local ring $R$ has an (essentially unique) minimal free resolution, and that when $R$ is regular, the length of this resolution is at most $\dim R$.

Let us prove the first assertion. It amounts to saying that $L^{-j}i^{\ast}_{x}\mathscr{F}=0$ implies $L^{-(j+1)}i^{\ast}_{x}\mathscr{F}=0$. The stalk $\mathscr{F}_{x}$ is a finitely generated module over the local ring $\mathscr{O}_{X,x}$, and in its minimal free resolution, the rank of the free $\mathscr{O}_{X,x}$-module in degree $-j$ is equal to $\dim L^{-j}i^{\ast}_{X}\mathscr{F}$. If $L^{-j}i^{\ast}_{X}\mathscr{F}=0$, then the minimal free resolution has length at most $j$, and this trivially implies that $L^{-(j+1)}i^{\ast}_{x}\mathscr{F}=0$.

Now let us prove the second assertion. Fix an irreducible component $Z$ of $S^{-j}(\mathscr{F})$. After localizing at the generic point of $Z$, we obtain a finitely generated module $\mathscr{F}_{Z}$ over the local ring $\mathscr{O}_{X,Z}$. By Serre’s theorem, $\mathscr{O}_{X,Z}$ is a regular local ring of dimension $c=\operatorname{codim}(Z,X)$, and so the minimal free resolution of $\mathscr{F}_{Z}$ has lengt at most $c$. Since $\mathscr{F}$ is coherent, there is a Zariski-open subset $U\subseteq X$, containing the generic point of $Z$, such that \mathscr{F}\big{|}_{U} has a free resolution of length at most $c$. This implies that $L^{-j}i^{\ast}_{x}\mathscr{F}=0$ for every $j>c$ and every $x\in U(k)$, and since $Z\subseteq S^{-j}(\mathscr{F})$, the conclusion is that $c\geq j$. ∎

28.

We finish our discussion of the symmetric Fourier-Mukai transform and its properties by proving Proposition 10.1 from the introduction.

Proof of Proposition 10.1.

In $D_{\mathit{coh}}^{\mathit{b}}(\mathscr{O}_{B})$, we have a distinguished triangle

[TABLE]

where $\tau_{\geq 1}\colon D_{\mathit{coh}}^{\mathit{b}}(\mathscr{O}_{B})\to D_{\mathit{coh}}^{\geq 1}(\mathscr{O}_{B})$ is one of the truncation functors for the standard t-structure. Applying $\operatorname{\mathsf{FM}}_{B}$, which is exact and contravariant, we obtain another distinguished triangle

[TABLE]

in $D_{\mathit{coh}}^{\mathit{b}}(\mathscr{O}_{B^{\vee}})$. The complex on the left belongs to $D_{\mathit{coh}}^{\leq-1}(\mathscr{O}_{B^{\vee}})$, due to Proposition 6.1; also $\operatorname{\mathsf{FM}}_{B}(\mathbf{R}f_{\ast}\mathscr{F})=\mathbf{L}\hat{f}^{\ast}\operatorname{\mathsf{FM}}_{A}(\mathscr{F})=\mathbf{L}\hat{f}^{\ast}\hat{\mathscr{F}}$ by Proposition 4.1. Going from the distinguished triangle to the long exact sequence in cohomology leads to

[TABLE]

due to the fact that $f_{\ast}\mathscr{F}$ is also a GV-sheaf. ∎

D. Applications

29.

Now let us turn to the applications to GV-sheaves and M-regular sheaves. We start by proving two simple lemmas. Recall the notation

[TABLE]

for objects $K_{1}\in D_{\mathit{coh}}^{\mathit{b}}(\mathscr{O}_{A})$ and $K_{2}\in D_{\mathit{coh}}^{\mathit{b}}(\mathscr{O}_{B})$.

Lemma 29.1**.**

We have a natural isomorphism of bifunctors

[TABLE]

from $D_{\mathit{coh}}^{\mathit{b}}(\mathscr{O}_{A})\times D_{\mathit{coh}}^{\mathit{b}}(\mathscr{O}_{B})$ to $D_{\mathit{coh}}^{\mathit{b}}(\mathscr{O}_{A^{\vee}})\times D_{\mathit{coh}}^{\mathit{b}}(\mathscr{O}_{B^{\vee}})$.

Proof.

This follows from the isomorphism $P_{A\times B}=p^{\ast}_{13}P_{A}\otimes p^{\ast}_{24}P_{B}$ on the abelian variety $A\times B\times A^{\vee}\times B^{\vee}$, by the usual base change arguments. ∎

30.

The following lemma shows the effect of tensoring by an ample line bundle.

Lemma 30.1**.**

Let $L$ be an ample line bundle on $A$, and $K\in D_{\mathit{coh}}^{\mathit{b}}(\mathscr{O}_{A})$. Then the $k$-vector space \operatorname{Hom}_{k}\bigl{(}H^{i}(A,L\otimes K),k\bigr{)} is isomorphic to a direct summand of

[TABLE]

where $\phi_{L}\colon A\to A^{\vee}$ is the homomorphism in (9.1).

Proof.

If we apply Proposition 4.1 to the morphism $p\colon A\to\operatorname{Spec}k$, we obtain

[TABLE]

Now $L\otimes K=\mathbf{L}\Delta^{\ast}(L\boxtimes K)$, and the diagonal homomorphism $\Delta\colon A\to A\times A$ is dual to the multiplication morphism $m\colon A^{\vee}\times A^{\vee}\to A^{\vee}$. Another application of Proposition 4.1, followed by Lemma 29.1, therefore gives

[TABLE]

due to the fact that $L$ is a GV-sheaf. In the rest of the proof, we use the following commutative diagram:

[TABLE]

Base change for the smooth morphism $m$ yields an isomorphism

[TABLE]

Since $\phi_{L}$ is an isogeny, \mathbf{R}p_{\ast}\bigl{(}i^{\ast}\hat{L}\otimes\operatorname{\mathsf{FM}}_{A}(K)\bigr{)} is isomorphic to a direct summand of

[TABLE]

Using Proposition 9.2, we can rewrite this in the form

[TABLE]

We now get the desired result by taking cohomology. ∎

31.

We can now prove Hacon’s criterion for GV-sheaves in terms of vanishing.

Proof of Theorem 11.1.

Suppose that $\mathscr{F}$ is a GV-sheaf on $A$. The pullback of $\mathscr{F}$ by an isogeny $\varphi\colon A^{\prime}\to A$ is again a GV-sheaf on $A^{\prime}$; this follows from the identities in Proposition 4.1 because

[TABLE]

Here we used the fact that isogenies are flat and finite. We can therefore assume without loss of generality that $\varphi=\operatorname{id}_{A}$. Now let $L$ be any ample line bundle on $A$. To prove the first half of the theorem, we need to show that

[TABLE]

Lemma 30.1 reduces the problem to showing that

[TABLE]

But this is obvious because $\operatorname{\mathsf{FM}}_{A}(\mathscr{F})$ is a sheaf (and $\phi_{L}$ is flat).

Next, we prove the converse. Suppose that $\mathscr{F}\in\operatorname{Coh}(\mathscr{O}_{A})$ has the property stated in the theorem. We need to explain why $\mathcal{H}^{i}\operatorname{\mathsf{FM}}_{A}(\mathscr{F})=0$ for every $i\neq 0$; in fact, by Proposition 6.1, it would be enough to consider only $i<0$. After choosing a sufficiently ample line bundle $L^{\prime}$ on $A^{\vee}$, the problem reduces to showing that

[TABLE]

Another application of Lemma 30.1, but with the roles of $A$ and $A^{\vee}$ interchanged, shows that it is enough to prove the vanishing

[TABLE]

where $\phi_{L^{\prime}}\colon A^{\vee}\to A$ is the isogeny determined by $L^{\prime}$. This is obvious for $i>0$, and for $i<0$, it is exactly the condition in the statement of the theorem. ∎

32.

In the remainder of the paper, we relate M-regularity to (continuous) global generation. The underlying idea is extremely simple. Suppose that $\mathscr{F}\in\operatorname{Coh}(\mathscr{O}_{A})$ is a GV-sheaf, and set $\hat{\mathscr{F}}=\operatorname{\mathsf{FM}}_{A}(\mathscr{F})\in\operatorname{Coh}(\mathscr{O}_{A^{\vee}})$. Let $S\subseteq A^{\vee}(k)$ be any set of closed points. A small calculation shows that the evaluation morphism

[TABLE]

corresponds, under the (contravariant!) functor $\operatorname{\mathsf{FM}}_{A}$, to the morphism

[TABLE]

here $k(\alpha)=(t_{\alpha}\circ e)_{\ast}\mathscr{O}_{\operatorname{Spec}k}$. Now suppose that $\mathscr{F}$ is M-regular, and that the set $S\subseteq A^{\vee}(k)$ is dense in the Zariski topology. Since $\hat{\mathscr{F}}$ is torsion-free, the morphism in (32.2) must be injective, and so we should expect the evaluation morphism in (32.1) to be surjective. The only problem is that infinite sums are not coherent (and infinite products are not even quasi-coherent), which means that our results about $\operatorname{\mathsf{FM}}_{A}$ do not apply. This can be dealt with by making a pointwise argument.

33.

We now turn the above idea into a rigorous proof.

Proposition 33.1**.**

Suppose that $\mathscr{F}\in\operatorname{Coh}(\mathscr{O}_{A})$ is M-regular. Let $S\subseteq A^{\vee}(k)$ be any subset that is dense in the Zariski topology. Then there is a finite subset $F\subseteq S$ such that the evaluation morphism

[TABLE]

is surjective.

Proof.

Let $a\in A(k)$ be any closed point. Consider the evaluation morphism

[TABLE]

To show that $\operatorname{ev}_{\alpha}$ is surjective in a neighborhood of a closed point $a\in A(k)$, it is enough (by Nakayama’s lemma) to show that the induced morphism

[TABLE]

is surjective. By Proposition 5.1, we have

[TABLE]

Combining this with the formula in (25.2), we get

[TABLE]

using the translation invariance of $P_{-a}$. It follows that the morphism in (33.2) is dual to the morphism

[TABLE]

The conclusion is that $\operatorname{ev}_{\alpha}$ is surjective in a neighborhood of a closed point $a\in A(k)$ if and only if the morphism in (33.3) is injective.

Now $H^{0}(A^{\vee},\hat{\mathscr{F}}\otimes P_{-a})$ is a finite-dimensional $k$-vector space. Since $\hat{\mathscr{F}}$ is a torsion-free coherent sheaf, and $S\subseteq A^{\vee}(k)$ is dense in the Zariski topology, we can therefore find a finite subset $S(a)\subseteq S$ such that the morphism

[TABLE]

is injective. It follows that the dual morphism

[TABLE]

is surjective, and hence that the evaluation morphism

[TABLE]

is surjective on a Zariski-open subset containing the given closed point $a\in A(k)$. Finitely many of these open subsets cover $A$; consequently, there is a finite subset $F\subseteq S$ with the property that

[TABLE]

is surjective. This completes the proof. ∎

34.

We can now show that M-regularity implies (continuous) global generation.

Proof of Theorem 12.1.

The first assertion follows immediately from Proposition 33.1. For the second one, let $S\subseteq A^{\vee}(k)$ be the set of all torsion points; this is certainly dense in the Zariski topology. By Proposition 33.1, there is a finite subset $F\subseteq S$ such that

[TABLE]

is surjective. Let $\varphi\colon A^{\prime}\to A$ be an isogeny with the property that $\hat{\varphi}(\alpha)=0$ for every $\alpha\in F$; for example, multiplication by the greatest common divisor of the orders of the points in $F$ will do. The induced morphism

[TABLE]

is then still surjective, and so $\varphi^{\ast}\mathscr{F}$ is globally generated. ∎

35.

In fact, the argument above leads to the following stronger result.

Theorem 35.1**.**

Suppose that $\mathscr{F}\in\operatorname{Coh}(\mathscr{O}_{A})$ is torsion-free. Let $S\subseteq A(k)$ be any subset that is dense in the Zariski topology. Then there is a finite subset $F\subseteq S$ such that the evaluation morphism

[TABLE]

is surjective.

Proof.

Let $a\in A(k)$ and $\alpha\in A^{\vee}(k)$ be any pair of closed points. We have

[TABLE]

using Proposition 4.1, Proposition 5.1, and the projection formula (for $t_{-\alpha}$). Our fixed trivializations for $(e\times\operatorname{id})^{\ast}P_{A}$ and $(\operatorname{id}\times e)^{\ast}P_{A}$ determine an isomorphism

[TABLE]

which we can use to rewrite the identity above in the form

[TABLE]

Here $P_{(-a,-\alpha)}$ is the fiber of the Poincaré bundle at $(-a,-\alpha)\in(A\times A^{\vee})(k)$. Taking cohomology in degree zero, we see that

[TABLE]

using the identity in (15.1) to replace $P_{(-a,-\alpha)}$. Similarly, we have

[TABLE]

and since $\operatorname{\mathsf{FM}}_{A}(\mathscr{F})\in D_{\mathit{coh}}^{\leq 0}(\mathscr{O}_{A^{\vee}})$ by Proposition 6.1, it follows that

[TABLE]

The conclusion is that that the evaluation morphism

[TABLE]

on $A$ is dual to the evaluation morphism

[TABLE]

on $A^{\vee}$. Note that the second morphism fits into a commutative diagram

[TABLE]

Now we can argue as before to prove the assertion.

Fix a closed point $\alpha\in A^{\vee}(k)$. Since $\mathscr{F}$ is torsion-free, and $S\subseteq A(k)$ is dense in the Zariski topology, there is a finite subset $S(\alpha)\subseteq S$ such that the morphism

[TABLE]

is injective. After dualizing, it follows that the morphism

[TABLE]

is surjective. By Nakayama’s lemma, this means that

[TABLE]

is surjective on a Zariski-open subset containing the closed point $\alpha\in A^{\vee}(k)$. Finitely many of these open subsets cover $A^{\vee}$; consequently, there is a finite subset $F\subseteq S$ with the property that

[TABLE]

is surjective. This finishes the proof. ∎