Generic vanishing theory via mixed Hodge modules

A. Introduction

The attempt to understand cohomology vanishing statements on irregular varieties in the absence of strong positivity has led to what is usually called generic vanishing theory. Perhaps the most famous result is the generic vanishing theorem of Green and Lazarsfeld [GL1], which in a weak form states that on a smooth complex projective variety $X$, the cohomology of a generic line bundle $L\in\operatorname{Pic}^{0}(X)$ vanishes in degrees less than $\dim a(X)$, where $a\colon X\to\operatorname{Alb}(X)$ denotes the Albanese mapping of $X$. This theorem and its variants have found a surprising number of applications, ranging from results about singularities of theta divisors [EL] to recent work on the birational geometry of irregular varieties, including a proof of Ueno’s conjecture [ChH].

One can consider the set of those line bundles for which the cohomology in a given degree does not vanish, and thanks to the work of many people, the structure of these sets is very well understood. This is more precisely the content of generic vanishing theory. Denoting, for any coherent sheaf $\mathcal{F}$ on $X$, by

the $i$-th cohomological support locus of $\mathcal{F}$, its main statements are the following:

One has ${\rm codim}~{}V^{i}(\omega_{X})\geq i-\dim X+\dim a(X)$ for all $i$ [GL1, GL2]. This implies the generic vanishing theorem via Serre duality.

The irreducible components of each $V^{i}(\omega_{X})$ are torsion translates of abelian subvarieties of $\operatorname{Pic}^{0}(X)$ [GL2, Arapura, Simpson2].

If $p_{2}:X\times\operatorname{Pic}^{0}(X)\rightarrow\operatorname{Pic}^{0}(X)$ is the second projection, and $P$ is a Poincaré line bundle on $X\times\operatorname{Pic}^{0}(X)$, then $\mathbf{R}{p_{2}}_{*}P$ is locally around each point quasi-isomorphic to a linear complex [GL2]. A precise version of this result is known to imply (L), except for the torsion statement, and based on this also (D).

Analogous results have been considered for the cohomology of local systems, replacing $\operatorname{Pic}^{0}(X)$ by ${\rm Char}(X)$, the algebraic group parametrizing rank one local systems [Arapura, Simpson, Simpson2]. New approaches and extensions for the theory on $\operatorname{Pic}^{0}(X)$ have been introduced more recently, for example in [CH, Hacon, PP]. On the other hand, important gaps have remained in our understanding of some of the most basic objects. For instance, while (L) is also known for the sheaf of holomorphic $p$-forms $\Omega_{X}^{p}$ with $p<n$, a good generic Nakano-type vanishing statement as in (D) has eluded previous efforts, despite several partial results [GL1, PP]. The same applies to the case of local systems of rank one, where the perhaps the even more interesting property (SL) has been missing as well.

In this paper, we answer those remaining questions, and at the same time recover the previous results of generic vanishing theory mentioned above (with the exception of the statement about torsion points, which is of a different nature) by enlarging the scope of the study to the class of filtered $\mathscr{D}$-modules associated to mixed Hodge modules on abelian varieties. In fact, there is a version of the Fourier-Mukai transform for $\mathscr{D}$-modules, introduced by Laumon [Laumon2] and Rothstein [Rothstein]; it takes $\mathscr{D}$-modules on an abelian variety to complexes of coherent sheaves on $A^{\natural}$, the moduli space of line bundles on $A$ with integrable connection. Our main results can be summarized briefly as describing the Fourier-Mukai transform of the trivial $\mathscr{D}$-module $\mathscr{O}_{X}$ on an irregular variety $X$.

Why mixed Hodge modules?

To motivate the introduction of mixed Hodge modules into the problem, let us briefly recall the very elegant proof of the generic vanishing theorem for $\omega_{X}$ discovered by Hacon [Hacon]. It goes as follows.

in the same range. Now the sheaves $R^{i}a_{\ast}\omega_{X}$ still satisfy a Kodaira-type vanishing theorem, and together with the special geometry of abelian varieties this implies after some work that the Fourier-Mukai transform of $R^{i}a_{\ast}\omega_{X}$ is the dual of a coherent sheaf $\mathscr{F}_{i}$ on $\widehat{A}$, which is to say that

The desired inequality for the codimension of the support becomes

which is now a consequence of a general theorem about regular local rings. This proves the dimension statement (D), and hence the generic vanishing theorem for topologically trivial line bundles.

One of the subjects of this paper is to use this framework in order to prove a generic vanishing theorem for general objects of Hodge-theoretic origin. The role of Kollár’s theorem is played by the decomposition theorem [BBD], or more precisely by its Hodge-theoretic version due to Morihiko Saito [Saito-MHM]. This is one main reason why mixed Hodge modules form a natural setting here. Another is the existence of a very general Kodaira-type vanishing theorem for mixed Hodge modules, again due to Saito, which becomes particularly useful on abelian varieties. This vanishing theorem allows us to generalize the second half of the proof above to any coherent sheaf of Hodge-theoretic origin on an abelian variety. Finally, in order to extract the relevant information about the sheaves $\Omega_{X}^{p}$ with $p<\dim X$, one needs a result by Laumon and Saito on the behavior of filtered $\mathscr{D}$-modules under direct images, which only works well in the case of $\mathscr{D}$-modules that underlie mixed Hodge modules.

The main results

Let us now give a summary of the results we obtain. There are essentially two parts: vanishing and dimension results, for which Hodge modules are crucially needed, and linearity results, which apply to certain Hodge modules, but for which the general theory of $\mathscr{D}$-modules and the harmonic theory of flat line bundles suffice in the proofs. The theory of mixed Hodge modules is reviewed in §5 below.

The starting point is a general Kodaira-type vanishing theorem for the graded pieces of the de Rham complex of a mixed Hodge module, proved by Saito. On an abelian variety $A$, this can be improved to a vanishing theorem for coherent sheaves of the form $\operatorname{gr}_{k}^{F}\mathcal{M}$, where $(\mathcal{M},F)$ is any filtered $\mathscr{D}$-module underlying a mixed Hodge module on $A$ (see Lemma 9.1 below). We use this observation to produce natural classes of perverse coherent sheaves [AB, Kashiwara] on the dual abelian variety $\widehat{A}$, and on the parameter space for Higgs line bundles $\widehat{A}\times H^{0}(A,\Omega_{A}^{1})$.

We first show that every mixed Hodge module on $A$ gives rise to a collection of sheaves satisfying the generic vanishing condition, or equivalently perverse coherent sheaves on $\widehat{A}$ with respect to the dual standard $t$-structure (reviewed in §7).

Consequently, its Fourier-Mukai transform $\mathbf{R}\Phi_{P}(\operatorname{gr}_{k}^{F}\mathcal{M})$ is a perverse coherent sheaf on $\widehat{A}$.

This uses Hacon’s general strategy, as in §2 above, and the correspondence established in [Popa, PP] between objects satisfying generic vanishing (or $\operatorname{GV}$-objects) and perverse coherent sheaves in the above sense.

In order to obtain a generic Nakano-type vanishing statement similar to (D), or statements for cohomological support loci of rank one local systems, we apply Theorem 3.1 to the direct image of the trivial Hodge module on an irregular variety under the Albanese map. Here our main tools are the decomposition theorem for Hodge modules [Saito-HM], extending the well-known result of [BBD], and a formula due to Laumon [Laumon] for the behavior of the associated graded objects under projective direct images (which is true for mixed Hodge modules).

Our main results in this direction are the following. Let $X$ be a smooth complex projective variety of dimension $n$, with nonzero irregularity $g=h^{1}(X,\mathscr{O}_{X})$. Let $a\colon X\to A=\operatorname{Alb}(X)$ be the Albanese map of $X$. Consider the defect of semismallness of the Albanese map $a\colon X\to A$, which is defined by the formula

Let $X$ be a smooth complex projective variety of dimension $n$. Then

Suppose that the Albanese map of $X$ is semismall. Then

Unlike in the case of $\omega_{X}$, it is not sufficient to assume that the Albanese map is generically finite over its image; this was already pointed out in [GL1]. Nevertheless, our method also recovers the stronger statement for $\omega_{X}$ [GL1] and its higher direct images [Hacon] (see the end of §10). This is due to the special properties of the first non-zero piece of the Hodge filtration on mixed Hodge modules, established by Saito. Note also that, since $\chi(\Omega_{X}^{p})=\chi(\Omega_{X}^{p}\otimes L)$ for any $L\in\operatorname{Pic}^{0}(X)$ by the deformation invariance of the Euler characteristic of a coherent sheaf, we have as a consequence the following extension of the fact that $\chi(\omega_{X})\geq 0$ for varieties of maximal Albanese dimension.

If the Albanese map of $X$ is semismall, then $(-1)^{n-p}\chi(\Omega_{X}^{p})\geq 0$.

Let $X$ be a smooth complex projective variety of dimension $n$. Then

To deduce this from the arguments leading to Theorem 3.2, we need to appeal to the structure results and the relationship with the space of Higgs bundles, proved by Simpson [Simpson, Simpson2] and Arapura [Arapura]; see §12.

While editing this paper, we learned of the very interesting preprint [KW] by T. Krämer and R. Weissauer, who prove vanishing theorems for perverse sheaves on abelian varieties. They also obtain a generic vanishing theorem for $\Omega_{X}^{p}$ and for rank one local systems, involving the same quantity $\delta(a)$ as in Theorem 3.2, but without precise codimension bounds for the cohomological support loci. Their methods are very different from ours.

Three additional theorems complete the picture, by describing in detail the Fourier-Mukai transform of the trivial $\mathscr{D}$-module $\mathscr{O}_{X}$; they include results of type (D), (L) and (SL) on the space of line bundles with integrable connection on $X$. Here it is important to consider two different kinds of Fourier-Mukai transforms, corresponding in Simpson’s terminology [Simpson2] to the Dolbeault realization (via Higgs bundles) and the de Rham realization (via line bundles with integrable connection) of ${\rm Char}(X)$.

where $S=\operatorname{Sym}V^{\ast}$, and the complex in brackets is placed in degrees $-n,\dotsc,0$, with differential induced by the evaluation morphism $\mathscr{O}_{X}\otimes V\to\Omega_{X}^{1}$. Since this is a complex of finitely generated graded modules over ${\rm Sym}~{}\mathscr{T}_{A}$, it naturally corresponds to a complex of coherent sheaves on cotangent bundle $T^{*}A=A\times V$, namely

Let $a\colon X\to A$ be the Albanese map of a smooth complex projective variety of dimension $n$, and let $p_{1}\colon X\times V\to X$ be the first projection.

where each $\mathscr{C}_{i,j}$ is a Cohen-Macaulay sheaf of dimension $\dim A$.

The support of each $\mathbf{R}\Phi_{P}\mathscr{C}_{i,j}$ is a finite union of torsion translates of triple tori in $\widehat{A}\times V$, subject to the inequalities

The dual objects \mathbf{R}\mathcal{H}\mathit{om}\bigl{(}\mathbf{R}\Phi_{P}\mathscr{C}_{i,j},\mathscr{O}_{\widehat{A}\times V}\bigr{)} also satisfy (ii).

The second and third theorems are best stated in terms of the generalized Fourier-Mukai transform for $\mathscr{D}$-modules on abelian varieties, introduced by Laumon [Laumon2] and Rothstein [Rothstein]. Their work gives an equivalence of categories

We review the construction, both algebraically and analytically, in §17 below. This Fourier-Mukai transform is the right context for a strong linearity result (SL) for the $\mathscr{D}$-module $\mathscr{O}_{X}$, extending the result for topologically trivial line bundles in [GL2].

placed in degrees $-n,\dotsc,n$, with differential given by the formula

As discussed in §23, every direct summand of a linear complex (in the derived category) is again quasi-isomorphic to a linear complex. It follows that all direct summands of $\mathbf{R}\Phi_{P^{\natural}}\mathscr{O}_{X}$ coming from the decomposition

have the same linearity property (see Corollary 17.3). Note also that using base change for local systems and the description of tangent cones to cohomological support loci as in [Libgober], via arguments as in[GL2]*§4 (which we will not repeat here), Theorem 3.7 gives another proof of the linearity to the cohomological support loci $\Sigma^{k}(X)$, i.e. for the statement of type (L).

Our proof of Theorem 3.7 relies on the harmonic theory for flat line bundles developed by Simpson [Simpson]. As in [GL2], the idea is that after pulling the complex $\mathbf{R}\Phi_{P^{\natural}}\mathscr{O}_{X}$ back to the universal covering space of $A^{\natural}$, one can use harmonic forms to construct a linear complex that is quasi-isomorphic to the pullback in a neighborhood of a given point. Additional technical difficulties arise however from the fact that the wedge product of two harmonic forms is typically no longer harmonic (which was true in the special case of $(0,q)$-forms needed in [GL2], and led to a natural quasi-isomorphism in that case). The new insight in this part of the paper is that a quasi-isomorphism can still be constructed by more involved analytic methods.

Thirdly, using Theorem 3.5, we are able to derive the following generic vanishing-type property of the Laumon-Rothstein transform of the $\mathscr{D}$-module $\mathscr{O}_{X}$.

For any smooth projective complex variety $X$ we have

This is best expressed in terms of the new $t$-structure alluded to above; see the equivalent formulation in Theorem 24.1. In particular, in the special case when the Albanese map is semismall, it follows that $\mathbf{R}\Phi_{P^{\natural}}\mathscr{O}_{X}$ is a perverse coherent sheaf. Via formal algebraic properties of this $t$-structure, we deduce the following vanishing statement for the higher direct images $R^{i}\Phi_{P^{\natural}}\mathscr{O}_{X}$. It is the analogue of the corresponding statement for the standard Fourier-Mukai transform of $\mathscr{O}_{X}$ conjectured by Green and Lazarsfeld, and proved by Hacon [Hacon] (and in the Kähler setting in [PP2]).

For any smooth projective complex variety $X$ we have

Along the lines of [LP], Theorem 3.7 and Corollary 3.9 can be combined to give, by means of the BGG correspondence, a bound on the complexity of the cohomology algebra

Let $X$ be a smooth projective complex variety of dimension $n$. Then $E$ is \bigl{(}n+\delta(a)\bigr{)}-regular as a graded $E$-module.

Concretely, this means that $P_{X}$ is generated in degrees $-\delta(a),\dotsc,n$; the relations among the generators appear in degrees $-\delta(a)-1,\dotsc,n-1$; and more generally, the $p$-th module of syzygies of $P_{X}$ has all its generators in degrees $-\delta(a)-p,\dotsc,n-p$. Simple examples show that this result is optimal; see §25.

The last part of the paper contains a few statements extending our results to more general classes of $\mathscr{D}$-modules on abelian varieties, not necessarily underlying mixed Hodge modules. The proofs are of a different nature, and will be presented elsewhere. We propose a few natural open problems as well.

Acknowledgements

We are grateful to Mark de Cataldo for pointing out the current statement of Proposition 11.2, which is a strengthening of our original result. We thank Donu Arapura, Takuro Mochizuki and Keiji Oguiso for useful conversations. M.P. is partially supported by NSF grant DMS-1101323. C.S. is partially supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and by NSF grant DMS-1100606.

B. Preliminaries

In this section, we recall a few aspects of Morihiko Saito’s theory of mixed Hodge modules [Saito-MHM] which, as explained in the introduction, offers a convenient setting for the results of this paper. A very good survey can also be found in [Saito-survey]. We only discuss the case of a complex algebraic variety $X$, say of dimension $n$.

is the analogue for Hodge modules of the decomposition theorem of [BBD]. The proof of this result with methods from algebraic analysis and $\mathscr{D}$-modules is one of the main achievements of Saito’s theory.

A regular holonomic $\mathscr{D}_{X}$-module $\mathcal{M}$, together with a good filtration $F_{\bullet}\mathcal{M}$ by $\mathscr{O}_{X}$-coherent subsheaves such that $\operatorname{gr}_{\bullet}^{F}\!\mathcal{M}$ is coherent over $\operatorname{gr}_{\bullet}^{F}\mathscr{D}_{X}$. This filtration plays the role of a Hodge filtration on $\mathcal{M}$.

These components are subject to several conditions, which are defined by induction on the dimension of the support of $\mathcal{M}$. On a one-point space, a mixed Hodge module is a graded-polarizable mixed Hodge structure; in general, Saito’s conditions require that the nearby and vanishing cycles of $\mathcal{M}$ with respect to any locally defined holomorphic function are again mixed Hodge modules (now on a variety of dimension $n-1$); the existence of polarizations; etc. An important fact is that these local conditions are preserved after taking direct images, and that they are sufficient to obtain global results such as the decomposition theorem.

Three theorems about mixed Hodge modules

In this section, we recall from the literature three useful theorems about the associated graded object $\operatorname{gr}_{\bullet}^{F}\mathcal{M}$, for a filtered $\mathscr{D}$-module $(\mathcal{M},F)$ underlying a mixed Hodge module.

During the discussion, $X$ will be a smooth complex projective variety of dimension $n$, and $M\in\operatorname{MHM}(X)$ a mixed Hodge module on $X$. As usual, we denote the underlying filtered $\mathscr{D}$-module by $(\mathcal{M},F)$. Recall that the associated graded of the sheaf of differential operators $\mathscr{D}_{X}$, with respect to the filtration by order of differential operators, is isomorphic to $\mathcal{A}_{X}^{\bullet}=\operatorname{Sym}^{\bullet}\mathscr{T}_{X}$, the symmetric algebra of the tangent sheaf of $X$. Since $\operatorname{gr}_{\bullet}^{F}\mathcal{M}$ is finitely generated over this sheaf of algebras, it defines a coherent sheaf $\mathscr{C}(\mathcal{M},F)$ on the cotangent bundle $T^{\ast}X$. The support of this sheaf is the characteristic variety of the $\mathscr{D}$-module $\mathcal{M}$, and therefore of pure dimension $n$ because $\mathcal{M}$ is holonomic.

The first of the three theorems is Saito’s generalization of the Kodaira vanishing theorem. Before we state it, observe that the filtration $F_{\bullet}\mathcal{M}$ is compatible with the $\mathscr{D}$-module structure on $\mathcal{M}$, and therefore induces a filtration on the de Rham complex of $\mathcal{M}$ by the formula

The associated graded complex for the filtration in (6.1) is

which is now a complex of coherent sheaves of $\mathscr{O}_{X}$-modules in degrees $-n,\dotsc,0$. This complex satisfies the following Kodaira-type vanishing theorem.

Let $(\mathcal{M},F)$ be the filtered $\mathscr{D}$-module underlying a mixed Hodge module on a smooth projective variety $X$, and let $L$ be any ample line bundle.

One has \mathbf{H}^{i}\bigl{(}X,\operatorname{gr}_{k}^{F}\operatorname{DR}_{X}(\mathcal{M})\otimes L\bigr{)}=0 for all $i>0$.

One has \mathbf{H}^{i}\bigl{(}X,\operatorname{gr}_{k}^{F}\operatorname{DR}_{X}(\mathcal{M})\otimes L^{-1}\bigr{)}=0 for all $i<0$.

The proof works by reducing the assertion to a vanishing theorem for perverse sheaves on affine varieties, with the help of Saito’s formalism. Details can be found in [Saito-MHM]*Proposition 2.33. ∎

The second theorem gives more information about the coherent sheaf $\mathscr{C}(\mathcal{M},F)$ on the cotangent bundle of $X$. Before stating it, we recall the definition of the Verdier dual $M^{\prime}=\mathbf{D}_{X}M$ of a mixed Hodge module. If $\operatorname{rat}M$ is the perverse sheaf underlying $M$, then $\operatorname{rat}M^{\prime}$ is simply the usual topological Verdier dual. On the level of $\mathscr{D}$-modules, note that since $\mathcal{M}$ is left $\mathscr{D}$-module, the dual complex

is naturally a complex of right $\mathscr{D}$-modules; since $\mathcal{M}$ is holonomic, it is quasi-isomorphic to a single right $\mathscr{D}$-module. If $\mathcal{M}^{\prime}$ denotes the left $\mathscr{D}$-module underlying the Verdier dual $M^{\prime}$, then that right $\mathscr{D}$-module is $\omega_{X}\otimes_{\mathscr{O}_{X}}\mathcal{M}^{\prime}$. Thus we have

where the right $\mathscr{D}$-module structure on $\omega_{X}\otimes\mathcal{M}^{\prime}$ is given by the rule $\xi\cdot(\omega\otimes m^{\prime})=(\xi\omega)\otimes m^{\prime}-\omega\otimes(\xi m^{\prime})$ for $\xi\in\mathscr{T}_{X}$. The Hodge filtration on $\mathcal{M}$ induces a filtration on the dual complex and hence on $\mathcal{M}^{\prime}$. The following result shows that this induced filtration is well-behaved, in a way that makes duality and passage to the associated graded compatible with each other.

Let $M$ be a mixed Hodge module on a smooth complex algebraic variety $X$ of dimension $n$, and let $M^{\prime}$ denote its Verdier dual. Then

where sections of $\operatorname{Sym}^{k}\mathscr{T}_{X}$ act with an extra factor of $(-1)^{k}$ on the right-hand side. If we consider both sides as coherent sheaves on $T^{\ast}X$, we obtain

where $p\colon T^{\ast}X\to X$ is the projection. In particular, $\mathscr{C}(\mathcal{M},F)$ is always a Cohen-Macaulay sheaf of dimension $n$.

That the coherent sheaf $\mathscr{C}(\mathcal{M},F)$ on $T^{\ast}X$ is Cohen-Macaulay is proved in [Saito-HM]*Lemme 5.1.13. For an explanation of how this fact implies the formula for the dual of the graded module $\operatorname{gr}_{\bullet}^{F}\mathcal{M}$, see [duality]*§3.1. ∎

The last of the three theorems gives a formula for the associated graded object of $f_{*}M$, where $f\colon X\to Y$ is a projective morphism between two smooth complex algebraic varieties. The formula itself first appears in a paper by Laumon [Laumon]*Construction 2.3.2, but it is not true in the generality claimed there (that is to say, for arbitrary filtered $\mathscr{D}$-modules).

Before stating the precise result, we give an informal version. The following diagram of morphisms, induced by $f$, will be used throughout:

Let $f\colon X\to Y$ be a projective morphism between smooth complex algebraic varieties, and let $M\in\operatorname{MHM}(X)$. Then with notation as above,

This can be easily proved using Saito’s formalism of induced $\mathscr{D}$-modules; both the factor of $\omega_{X/Y}$ and the shift in the grading come from the transformation between left and right $\mathscr{D}$-modules that is involved. To illustrate what is going on, we shall outline a proof based on factoring $f$ through its graph. By this device, it suffices to verify the formula in two cases: (1) for a regular closed embedding $f\colon X\hookrightarrow Y$; (2) for a projection $f\colon Y\times Z\to Y$ with $Z$ smooth and projective.

and so the formula we need to prove is that

By [Saito-HM]*p. 850, we have $f_{\ast}(\mathcal{M},F)\simeq\mathcal{M}[\partial_{1},\dotsc,\partial_{r}]$, with filtration given by

which is the desired formula. Globally, the factor of $\det N_{X\mid Y}$ is needed to make the above isomorphism coordinate independent.

Next, consider the case where $X=Y\times Z$, with $Z$ smooth and projective of dimension $r$, and $f=p_{1}$. Then $\omega_{X/Y}\simeq p_{2}^{\ast}\omega_{Z}$. In this case, we have

where $\operatorname{DR}_{Y\times Z/Y}(\mathcal{M})$ is the relative de Rham complex

supported in degrees $-r,\dotsc,0$. As in (6.1), the Hodge filtration on the $\mathscr{D}$-module $\mathcal{M}$ induces a filtration on the relative de Rham complex by the formula

Now the key point is that since $Z$ is smooth and projective, the induced filtration on the direct image complex is strict by [Saito-MHM]*Theorem 2.14. It follows that

On the other hand, a graded locally free resolution of $\omega_{Z}$ as a graded $\mathcal{A}_{Z}$-module is given by the complex

again in degrees $-r,\dotsc,0$. Therefore $\omega_{X/Y}\otimes_{\mathscr{O}_{X}}f^{\ast}\mathcal{A}_{Y}$ is naturally resolved, as a graded $\mathcal{A}_{X}$-module, by the complex

and so $\operatorname{gr}_{\bullet+r}^{F}\mathcal{M}\otimes_{\mathcal{A}_{X}}f^{\ast}\mathcal{A}_{Y}\otimes_{\mathscr{O}_{X}}\omega_{X/Y}$ is represented by the complex

from which the desired formula follows immediately. ∎

Perverse coherent sheaves

The following fundamental result is proved in [Kashiwara]*Theorem 5.9 and, based on an idea of Deligne, in [AB]*Theorem 3.10.

which means that the duality functor $\mathbf{R}\mathcal{H}\mathit{om}(-,\mathscr{O}_{X})$ exchanges the two perverse $t$-structures defined by $p$ and $\hat{p}$. The heart of the $t$-structure defined by $p$ is denoted

and is called the abelian category of $p$-perverse coherent sheaves.

The dual object $\mathbf{R}\mathcal{H}\mathit{om}(E,\mathscr{O}_{X})$ is a coherent sheaf.

It is easily verified that both $m$ and the dual function

The next lemma follows easily from [Kashiwara]*Lemma 5.5.

The perverse $t$-structures defined by $m$ and $\hat{m}$ satisfy

By duality, this also describes the subcategories with $\geq k$.

By translation it is enough to prove this for $k=0$, where the result is obvious from the fact that $m(x)\geq 0$. ∎

Integral functors and GV-objects

(2) For any sufficiently ample line bundle $L$ on $Y$, and every $i>k$,

(3) $R^{i}\Phi_{P^{\vee}}(\omega_{X}\otimes E^{\vee})=0$ for all $i<n-k$.

An object $E$ satisfying the equivalent conditions in the Theorem with $k=0$, is called a $\operatorname{GV}$-object (with respect to $P$); if it is moreover a sheaf, then it is called a $\operatorname{GV}$-sheaf.

C. Mixed Hodge modules and generic vanishing

Let $A$ be a complex abelian variety of dimension $g$, and let $M\in\operatorname{MHM}(A)$ be a mixed Hodge module on $A$. As usual, we denote the underlying filtered holonomic $\mathscr{D}$-module by $(\mathcal{M},F)$. From Theorem 6.2, we know that the associated graded pieces of the de Rham complex $\operatorname{DR}(\mathcal{M},F)$ satisfy an analogue of Kodaira vanishing. A key observation is that on abelian varieties, the same vanishing theorem holds for the individual coherent sheaves $\operatorname{gr}_{k}^{F}\mathcal{M}$, due to the fact that the cotangent bundle of $A$ is trivial. As we shall see, this implies that each $\operatorname{gr}_{k}^{F}\mathcal{M}$ is a $\operatorname{GV}$-sheaf on $A$ (and therefore transforms to a perverse coherent sheaf on $\widehat{A}$ with respect to the $t$-structure given by $c$ in §7).

supported in degrees $-g,\dotsc,0$. According to Theorem 6.2, this complex has the property that, for $i>0$,

Using the fact that $\Omega_{A}^{1}\simeq\mathscr{O}_{A}^{\oplus g}$, one can deduce the asserted vanishing theorem for the individual sheaves $\operatorname{gr}_{k}^{F}\mathcal{M}$ by induction on $k$. Indeed, since $\operatorname{gr}_{k}^{F}\mathcal{M}=0$ for $k\ll 0$, inductively one has for each $k$ a distinguished triangle

with $E_{k}$ an object satisfying $\mathbf{H}^{i}(A,E_{k}\otimes L)=0$. ∎

We now obtain the first theorem of the introduction, by combining Lemma 9.1 with Theorem 8.1 and a trick invented by Mukai. As mentioned above, the method is the same as in Hacon’s proof of the generic vanishing theorem [Hacon, PP].

Let $L$ be an ample line bundle on $\widehat{A}$. By Theorem 8.1, it suffices to show that

for $i>0$. Let $\varphi_{L}\colon\widehat{A}\to A$ be the isogeny induced by $L$. Then, by virtue of $\varphi_{L}$ being étale,

is injective, and so we are reduced to proving that the group on the right vanishes whenever $i>0$.

Let $N=\varphi_{L}^{\ast}M$ be the pullback of the mixed Hodge module $M$ to $\widehat{A}$. If $(\mathcal{N},F)$ denotes the underlying filtered holonomic $\mathscr{D}$-module, then $F_{k}\mathcal{N}=\varphi_{L}^{\ast}F_{k}\mathcal{M}$ because $\varphi_{L}$ is étale. On the other hand, by [Mukai] 3.11

Going back to the de Rham complex, it is worth recording the complete information one can obtain about $\operatorname{gr}_{k}^{F}\operatorname{DR}_{A}(\mathcal{M})$ from Saito’s theorem, since this produces further natural examples of $\operatorname{GV}$-objects on $A$. One one hand, just as in the proof of Theorem 3.1, we see that each $\operatorname{gr}_{k}^{F}\operatorname{DR}_{A}(\mathcal{M})$ is a $\operatorname{GV}$-object. On the other hand, since $\operatorname{gr}_{k}^{F}\operatorname{DR}_{A}(\mathcal{M})$ is supported in non-positive degrees, its transform with respect to the standard Fourier-Mukai functor

given by the normalized Poincaré bundle on $A\times\widehat{A}$, could a priori have cohomology in negative degrees. The following proposition shows that this is not the case.

If $(\mathcal{M},F)$ underlies a mixed Hodge module on $A$, then

By a standard application of Serre vanishing (see [PP]*Lemma 2.5), it suffices to show that for any sufficiently positive ample line bundle $L$ on $\widehat{A}$,

for $i<0$. Assuming that $L$ is symmetric, $R^{0}\Psi_{P}(L)$ is easily seen to be the dual of the locally free sheaf $R^{g}\Psi_{P}(L^{-1})$, and so arguing as in the proof of Theorem 3.1, we are reduced to proving that

whenever $i<0$. But since $\varphi_{L}^{\ast}\operatorname{gr}_{k}^{F}\operatorname{DR}(\mathcal{M},F)\simeq\operatorname{gr}_{k}^{F}\operatorname{DR}(\mathcal{N},F)$, this is an immediate consequence of part (2) of Saito’s vanishing Theorem 6.2. ∎

If $(\mathcal{M},F)$ underlies a mixed Hodge module on $A$, then

are $\operatorname{GV}$-objects on $A$. Therefore the graded pieces of the de Rham complexes associated to such $\mathscr{D}$-modules form a class of $\operatorname{GV}$-objects which is closed under Grothendieck duality.

The decomposition theorem for the Albanese map

Let $X$ be a smooth complex projective variety of dimension $n$, let $A=\operatorname{Alb}(X)$ be its Albanese variety, and let $a\colon X\to A$ be the Albanese map (for some choice of base point). As before, we set $g=\dim A$.

where $M_{i,j}$ has strict support equal to some irreducible subvariety $Z_{i,j}\subseteq A$; the perverse sheaf underlying $M_{i,j}$ is the intersection complex of a local system on a Zariski-open subset of $Z_{i,j}$. Note that since $a$ is projective, we have the Lefschetz isomorphism [Saito-MHM]*Théorème 1

induced by $i$-fold cup product with the first Chern class of an ample line bundle. The Tate twist, necessary to change the weight of $M_{i}$ from $n+i$ to $n-i$, requires some explanation. If $(\mathcal{M}_{i},F)$ denotes the filtered $\mathscr{D}$-module underlying $M_{i}$, then the filtered $\mathscr{D}$-module underlying $M_{i}(i)$ is $(\mathcal{M}_{i},F_{\bullet-i})$; thus the above isomorphism means that $F_{k}\mathcal{M}_{-i}\simeq F_{k-i}\mathcal{M}_{i}$.

with differential induced by the evaluation morphism $\mathscr{O}_{X}\otimes H^{0}(X,\Omega_{X}^{1})\to\Omega_{X}^{1}$.

The cotangent bundle of $A$ is isomorphic to the product $A\times V$, and so using the notation from §6, we have $\mathcal{A}_{A}=\mathscr{O}_{A}\otimes S$ as well as $a^{*}\mathcal{A}_{A}=\mathscr{O}_{X}\otimes S$. Consequently, $\omega_{X}\otimes a^{*}\mathcal{A}_{A}$ can be resolved by the complex

placed as usual in degrees $-n,\dotsc,0$. Applying Theorem 6.5, we find that

with Koszul-type differential induced by the morphism $\mathscr{O}_{X}\to\Omega_{X}^{1}\otimes V^{\ast}$, which in turn is induced by the evaluation morphism $\mathscr{O}_{X}\otimes V\to\Omega_{X}^{1}$. ∎

On the other hand, we know from the discussion above that

It follows that $\operatorname{gr}_{\bullet}^{F}a_{*}(\mathscr{O}_{X},F)$ splits, as a complex of graded modules over $\mathcal{A}_{A}=\mathscr{O}_{A}\otimes S$, into a direct sum of modules of the form $\operatorname{gr}_{\bullet}^{F}\mathcal{M}_{i,j}$. Putting everything together, we obtain a key isomorphism which relates generic vanishing, zero sets of holomorphic one-forms on $X$, and the topology of the Albanese mapping.

in the bounded derived category of graded $\mathscr{O}_{A}\otimes S$-modules.

Since the $M_{i,j}$ are Hodge modules on an abelian variety, it follows from Theorem 3.1 that each $\operatorname{gr}_{k}^{F}\mathcal{M}_{i,j}$ is a $\operatorname{GV}$-sheaf on $A$. The isomorphism in Corollary 10.2 shows that whether or not the entire complex $\operatorname{gr}_{k}^{F}a_{\ast}(\mathscr{O}_{X},F)$ satisfies a generic vanishing theorem is determined by the presence of nonzero $M_{i,j}$ with $i>0$. We shall see in §12 how this leads to a generic vanishing theorem of Nakano-type.

The defect of semismallness

The defect of semismallness of the map $a\colon X\to A$ is

is the usual mixed Hodge structure on the cohomology of the projective variety $F$. By the above, this mixed Hodge structure is nonzero; it follows for dimension reasons that $n-\dim Z+k\leq 2\dim F$. Consequently, $k\leq\delta(a)$ as claimed.

Now the conditions of support for the perverse sheaf $\operatorname{rat}M_{i}$ imply that

The final assertion is a consequence of the isomorphism $M_{-k}\simeq M_{k}(k)$. ∎

Generic vanishing on the Picard variety

We now address the generic vanishing theorem of Nakano type, Theorem 3.2 in the introduction, and related questions. Ideally, such a theorem would say that

has codimension at least $\lvert p+q-n\rvert$ in $\widehat{A}$. Unfortunately, such a good statement is not true in general (see Example 12.3 below, following [GL1]). To simplify our discussion of what the correct statement is, we make the following definition.

Let $X$ be a smooth projective variety. We say that $X$ satisfies the generic Nakano vanishing theorem with index $k$ if

Note that the absolute value is consistent with Serre duality, which implies that

We can use the analysis in §10 to obtain a precise formula for the index $k$, namely we show that $X$ satisfies generic Nakano vanishing with index $\delta(a)$, but not with index $\delta(a)-1$.

Recall from Proposition 11.2 that we have

It follows from the equivalence established in [Popa]*§3 that generic Nakano vanishing with index $k$ is equivalent to the statement that

We shall prove that this formula holds with $k=\delta(a)$ by descending induction on $p\geq 0$, starting from the trivial case $p=n+1$. To simplify the bookkeeping, we set

Our result explains the original counterexample from [GL1]*§3. The example consisted in blowing up an abelian variety $A$ of dimension four along a smooth curve $C$ of genus at least two; if $X$ denotes the resulting variety, then $a\colon X\to A$ is the Albanese mapping, and a short computation shows that

for every $L\in\operatorname{Pic}^{0}(A)$. This example makes it clear that the index in the generic Nakano vanishing theorem is not equal to the dimension of the generic fiber of the Albanese mapping. On the other hand, it is not hard to convince oneself that

Thus $\delta(a)=1$ is the correct value for the index in this case.

Theorem 3.2, combined with the equivalence between (1) and (3) in Theorem 8.1, implies a vanishing result for the cohomology sheaves of the Fourier-Mukai transform of any $\Omega_{X}^{p}$, in analogy with the result for $\mathscr{O}_{X}$ conjectured by Green and Lazarsfeld and proved in [Hacon] (note again that $P^{-1}\simeq(1\times(-1))^{*}P$, so for vanishing $P$ and $P^{-1}$ can be used interchangeably).

With the notation above, for every integer $p$ we have

Moreover, $R^{n-p-\delta(a)}\Phi_{P}\Omega_{X}^{p}\neq 0$ for some $p$.

Let us finally note that Theorem 3.2 and its proof improve the previously known generic vanishing results for $\Omega_{X}^{p}$ with $p<n$, and recover those for $\omega_{X}$ (or its higher direct images):

First, it was proved in [PP]*Theorem 5.11 that

with $\mu(a)={\rm max}\{k,m-1\}$, $k$ being the minimal dimension and $m$ the maximal dimension of a fiber of the Albanese map. It is a routine check that $\delta(a)\leq\mu(a)$.

Secondly, in the case of the lowest nonzero piece of the filtration on $a_{*}(\mathscr{O}_{X},F)$ the situation is better than what comes out of Theorem 3.2. This allows one to recover the original generic vanishing theorem for $\omega_{X}$ of [GL1], as well as its extension to higher direct images $R^{j}a_{*}\omega_{X}$ given in [Hacon]. Indeed, in the proof above note that

This shows that $R^{i}a_{*}\omega_{X}\simeq\operatorname{gr}_{g-n}^{F}\mathcal{M}_{i}$. Since these sheaves are torsion-free by virtue of Kollár’s theorem, it follows that $\operatorname{gr}_{g-n}^{F}\mathcal{M}_{i}=0$ unless $0\leq i\leq\dim X-\dim a(X)$. Thus one recovers the original generic vanishing theorem of Green and Lazarsfeld. A similar argument works replacing $\omega_{X}$ by higher direct images $R^{i}f_{*}\omega_{Y}$, where $f\colon Y\rightarrow X$ is a projective morphism with $Y$ smooth and $X$ projective and generically finite over $A$.

D. Cohomological support loci for local systems

Let $A$ be an abelian variety of dimension $g$, and set as before $V=H^{0}(A,\Omega_{A}^{1})$ and $S=\operatorname{Sym}V^{\ast}$. Let $M$ be a mixed Hodge module on $A$, with underlying filtered $\mathscr{D}$-module $(\mathcal{M},F)$. Then $\operatorname{gr}_{\bullet}^{F}\mathcal{M}$ is a finitely-generated graded module over $\operatorname{Sym}\mathscr{T}_{A}\simeq\mathscr{O}_{A}\otimes S$, and we denote the associated coherent sheaf on $T^{\ast}A=A\times V$ by $\mathscr{C}(\mathcal{M},F)$. We may then define the total Fourier-Mukai transform of $\operatorname{gr}_{\bullet}^{F}\mathcal{M}$ to be

where the notation is as in the following diagram:

Let now $X$ be a smooth projective variety of dimension $n$, let $a\colon X\to A$ its Albanese map, and consider again the decomposition

Denote by $\mathscr{C}_{i,j}=\mathscr{C}(\mathcal{M}_{i,j},F)$ the coherent sheaf on $T^{\ast}A=A\times V$ determined by the Hodge module $M_{i,j}$. The supports in $\widehat{A}\times V$ of the total Fourier-Mukai transforms $\mathbf{R}\Phi_{P}\mathscr{C}_{i,j}$ are of a very special kind; this follows by using a result of Arapura [Arapura]. To state the result, we recall the following term coined by Simpson [Simpson2]*p. 365.

A triple torus is any subvariety of $\widehat{A}\times H^{0}(A,\Omega_{A}^{1})$ of the form

for a surjective morphism $\varphi\colon A\to B$ to another abelian variety $B$. A subvariety is called a torsion translate of a triple torus if it is a translate of a triple torus by a point of finite order in $A\times H^{0}(A,\Omega_{A}^{1})$.

With notation as above, every irreducible component of the support of $\mathbf{R}\Phi_{P}\mathscr{C}_{i,j}$ is a torsion translate of a triple torus in $\widehat{A}\times V$.

where $p_{1}\colon X\times V\to X$, and the complex in brackets is placed in degrees $-n,\dotsc,0$, and has differential induced by the evaluation map $\mathscr{O}_{X}\otimes V\to\Omega_{X}^{1}$.

where we denote the Koszul complex associated to a single one-form $\omega\in V$ by

It follows from [Arapura]*Corollary on p. 312 and [Simpson2] that each irreducible component of $Z^{k}(m)$ is a torsion translate of a triple torus. To relate this information to the support of the total Fourier-Mukai transforms $\mathbf{R}\Phi_{P}\mathscr{C}_{i,j}$, we also introduce

Finally, we observe that each irreducible component of \operatorname{Supp}\bigl{(}\mathbf{R}\Phi_{P}\mathscr{C}_{i,j}\bigr{)} must also be an irreducible component of one the sets $Z_{i,j}^{k}:=Z_{i,j}^{k}(1)$, which concludes the proof. More precisely, we have

Indeed, the base change theorem shows that \operatorname{Supp}\bigl{(}R^{k}\Phi_{P}\mathscr{C}_{i,j}\bigr{)}\subset Z^{k}_{i,j} for all $k$, with equality if $k\gg 0$. Now assume that $(L,\omega)\in Z^{k}_{i,j}$ is a general point of a component which is not contained in \operatorname{Supp}\bigl{(}R^{k}\Phi_{P}\mathscr{C}_{i,j}\bigr{)}. We claim that then $(L,\omega)\in Z^{k+1}_{i,j}$, which concludes the proof by descending induction. If this were not the case, then again by the base change theorem, the natural map

would be surjective, which would contradict (L,\omega)\notin\operatorname{Supp}\bigl{(}R^{k}\Phi_{P}\mathscr{C}_{i,j}\bigr{)}. ∎

Generic vanishing for rank one local systems

One can define an isomorphism of real (but not complex) algebraic Lie groups

where the Koszul-type complex in brackets is again placed in degrees $-n,\dotsc,0$.

We can now obtain the generic vanishing theorem for local systems of rank one, stated in the introduction.

Note first that, since Verdier duality gives an isomorphism

we only need to prove the asserted inequality for $k\geq n$. Furthermore, Lemma 14.1 shows that it is enough to prove, for $k\geq 0$, the analogous inequality

for the subsets $Z^{k}(1)\subseteq{\rm Higgs}(X)$ that were introduced during the proof of Proposition 13.2. Recall from there that

Finally, we will see in a moment in the proof of Theorem 15.2 that

Duality and perversity for total transforms

In this section, we take a closer look at the behavior of the object $\mathscr{C}(\mathcal{M},F)$ under the total Fourier-Mukai transform. We begin with a result that shows how the total Fourier-Mukai transform interacts with Verdier duality for mixed Hodge modules.

Let $M$ be a mixed Hodge module on $A$, let $M^{\prime}$ be its Verdier dual, and let $\mathscr{C}(\mathcal{M},F)$ and $\mathscr{C}(\mathcal{M}^{\prime},F)$ be the associated coherent sheaves on $A\times V$. Then

where $\iota=(-1_{\widehat{A}})\times(-1_{V})$.

Recall that the Grothendieck dual on a smooth algebraic variety $X$ is given by \mathbf{D}_{X}(-)=\mathbf{R}\mathcal{H}\mathit{om}\bigl{(}-,\omega_{X}[\dim X]\bigr{)}. To simplify the notation, set $\mathscr{C}=\mathscr{C}(\mathcal{M},F)$ and $\mathscr{C}^{\prime}=\mathscr{C}(\mathcal{M}^{\prime},F)$. Then

For the first isomorphism we use Grothendieck duality, while for the second we use Theorem 6.4. Since $\dim\widehat{A}\times V=2g$, this implies the result. ∎

With notation as above, each $\mathbf{R}\Phi_{P}\mathscr{C}_{i,j}$ is a $m$-perverse coherent sheaf on $\widehat{A}\times V$. More precisely, the support of the object $\mathbf{R}\Phi_{P}\mathscr{C}_{i,j}$ is a finite union of torsion translates of triple tori, subject to the inequality

Let $q\colon\widehat{A}\times V\to\widehat{A}$ be the first projection. Since we are dealing with sheaves of graded modules, the support of the quasi-coherent sheaf

Perverse coherent sheaves on the space of holomorphic one-forms

We conclude this part by observing that, in analogy with the method described in [Popa], our method also produces natural perverse coherent sheaves on the affine space $V=H^{0}(A,\Omega_{A}^{1})$, where $A$ is an abelian variety of dimension $g$. We shall use the following projection maps:

This is an exercise in interchanging Grothendieck duality with pushforward by proper maps and pullback by smooth maps. ∎

We can essentially rephrase Lemma 9.1 as follows.

Let $M$ be a mixed Hodge module on an abelian variety $A$, with underlying filtered $\mathscr{D}$-module $(\mathcal{M},F)$, and let $\mathscr{C}(\mathcal{M},F)$ be the coherent sheaf on $A\times V$ associated to $\operatorname{gr}_{\bullet}^{F}\mathcal{M}$. Then for every ample line bundle $L$ on $A$, one has

$V$ being affine, it suffices to prove that the hypercohomology of the complex is concentrated in degree . But this hypercohomology is equal to

which vanishes for $i>0$ because of Lemma 9.1. ∎

We can now obtain perverse coherent sheaves on the affine space $V$ by pushing forward along the projection $q\colon A\times V\to V$.

Let $M$ be a mixed Hodge module on $A$, with underlying filtered $\mathscr{D}$-module $(\mathcal{M},F)$, and let $\mathscr{C}(\mathcal{M},F)$ be the coherent sheaf on $A\times V$ associated to $\operatorname{gr}_{\bullet}^{F}\mathcal{M}$. Then for every ample line bundle $L$ on $\widehat{A}$, one has

By Lemma 16.1 and Theorem 6.4, this object is isomorphic to

where $\mathscr{C}(\mathcal{M}^{\prime},F)$ is associated to the Verdier dual $M^{\prime}=\mathbf{D}_{A}M$. Now we apply the usual covering trick. Let $\varphi_{L}\colon\widehat{A}\to A$ be the isogeny defined by $L$. Then the object in (16.4) will belong to $\operatorname{Coh}(\mathscr{O}_{V})$ provided the same is true for

Since $\varphi_{L}^{\ast}\mathscr{C}(\mathcal{M}^{\prime},F)$ comes from the mixed Hodge module $\varphi_{L}^{\ast}M^{\prime}$, this is a consequence of Lemma 16.2. ∎

E. Strong linearity

We start by setting up some notation. Let $A$ be a complex abelian variety of dimension $g$, and let $A^{\natural}$ be the moduli space of algebraic line bundles with integrable connection on $A$. Note that $A^{\natural}$ naturally has the structure of a quasi-projective algebraic variety: on $\widehat{A}$, there is a canonical vector bundle extension

and $A^{\natural}$ is isomorphic to the preimage of $\widehat{A}\times\{1\}$ inside $E_{\widehat{A}}$. The projection

is thus a torsor for the trivial bundle $\widehat{A}\times H^{0}(A,\Omega_{A}^{1})$; this corresponds to the fact that $\nabla+\omega$ is again an integrable connection for any $\omega\in H^{0}(A,\Omega_{A}^{1})$. Note that $A^{\natural}$ is a group under tensor product, and that the trivial line bundle $(\mathscr{O}_{A},d)$ plays the role of the zero element.

Recall now that Laumon [Laumon2] and Rothstein [Rothstein] have extended the Fourier-Mukai transform to $\mathscr{D}$-modules. Their generalized Fourier-Mukai transform takes bounded complexes of coherent algebraic $\mathscr{D}$-modules on $A$ to bounded complexes of algebraic coherent sheaves on $A^{\natural}$; we briefly describe it following the presentation in [Laumon2]*§3, which is more convenient for our purpose. On the product $A\times A^{\natural}$, the pullback $P^{\natural}$ of the Poincaré bundle $P$ is endowed with a universal integrable connection $\nabla^{\natural}\colon P^{\natural}\to\Omega_{A\times A^{\natural}/A^{\natural}}^{1}\otimes P^{\natural}$, relative to $A^{\natural}$. Given any algebraic left $\mathcal{D}$-module $\mathcal{M}$ on $A$, interpreted as a quasi-coherent sheaf with integrable connection, we consider $p_{1}^{*}\mathcal{M}\otimes P^{\natural}$ on $A\times A^{\natural}$, endowed with the natural tensor product integrable connection $\nabla$ relative to $A^{\natural}$. We then define

where ${\rm DR}(p_{1}^{*}\mathcal{M}\otimes P^{\natural},\nabla)$ is the usual (relative) de Rham complex

placed in degrees $-g,\ldots,0$. As all of the entries in this complex are relative to $A^{\natural}$, it follows that $\mathbf{R}\Phi_{P^{\natural}}\mathcal{M}$ is represented by a complex of algebraic quasi-coherent sheaves on $A^{\natural}$. Restricted to coherent $\mathcal{D}$-modules, this is shown to induce an equivalence of categories

The same result is obtained by a different method in [Rothstein]*Theorem 6.2.

Since $A^{\natural}$ is not compact, it is essential to consider algebraic coherent sheaves on $A^{\natural}$ in the above equivalence. On $A$, this problem does not arise, because the category of coherent analytic $\mathscr{D}$-modules on a smooth projective variety is equivalent to the category of coherent algebraic $\mathscr{D}$-modules by a version of the GAGA theorem.

Now let $X$ be a smooth projective variety with Albanese map $a\colon X\to A$. By first pushing forward to $A$, or equivalently by working with the pullback of $(P^{\natural},\nabla^{\natural})$ to $X\times A^{\natural}$, one can similarly define

In this and the following six subsections, our goal is to prove the following linearity property for the Fourier-Mukai transform of the trivial $\mathscr{D}$-module $\mathscr{O}_{X}$ (see Definition 23.1 below for the definition of a linear complex over a local ring).

We conclude this subsection by noting that, regardless of the explicit linear representation, in combination with Proposition 23.3 below we obtain that every direct summand of $\mathbf{R}\Phi_{P^{\natural}}\mathscr{O}_{X}$ is also isomorphic to a linear complex in an analytic neighborhood of any given point on $A^{\natural}$.

Analytic description

and this is compatible with the exact sequence

We can similarly interpret the pullback of the Poincaré bundle to the complex manifold $X\times W$. Let $\mathscr{C}^{k}(X\times W/W)$ be the sheaf of smooth complex-valued relative $k$-forms on $X\times W$ that are in the kernel of $\bar{\partial}_{W}$, meaning holomorphic in the direction of $W$, and denote by

This leads to the following analytic description of the Fourier-Mukai transform (similar to [GL2]*Proposition 2.3). Let

be the differential operator defined by the rule $D(\alpha)=d_{X}\alpha+T\wedge\alpha$. Using that $d_{X}T=0$, it is easy to see that $D\circ D=0$.

The complex of $\mathscr{O}_{W}$-modules \bigl{(}C^{\bullet}(X\times W/W),D\bigr{)} is quasi-isomorphic to the pullback $\pi^{\ast}\mathbf{R}\Phi_{P^{\natural}}\mathscr{O}_{X}$, where $\pi:W\rightarrow A^{\natural}$ is the universal cover.

By the definition of the Fourier transform, $\mathbf{R}\Phi_{P^{\natural}}\mathscr{O}_{X}$ is the derived pushforward, via the projection $p_{2}\colon X\times A^{\natural}\to A^{\natural}$, of the complex

where $(P^{\natural},\nabla^{\natural})$ denotes the pullback of the Poincaré bundle to $X\times A^{\natural}$. Since $\pi\colon W\to A^{\natural}$ is a covering map, we thus obtain

To obtain the desired result, it suffices then to note that

This follows from a standard partition of unity argument as in [GH]*p. 42. ∎

Then the $R$-module $C^{k}$ consists of all convergent power series of the form

Then each differential $D\colon C^{k}\to C^{k+1}$ is given by the formula

Note that each $R$-module in the complex has infinite rank; moreover, in the formula for the differential $D$, the first of the two terms is not linear in $z_{1},\dotsc,z_{2g}$. Our goal is then to build a linear complex quasi-isomorphic to

by using harmonic forms with coefficients in the flat line bundle corresponding to $\tau$. The space of $d_{\tau}$-harmonic forms has the advantage of being finite-dimensional; in addition, any such form $\alpha\in A^{k}(X)$ satisfies $d_{\tau}\alpha=0$, and hence

This shows that the differential is linear when restricted to the free $R$-module generated by the $d_{\tau}$-harmonic forms. The only problem is that we do not obtain a subcomplex of $(C^{\bullet},D)$ in this way, because the wedge product $e_{j}\wedge\alpha$ is in general no longer harmonic. This difficulty can be overcome by constructing a more careful embedding of the space of $d_{\tau}$-harmonic $k$-forms into $C^{k}$, as we now explain.

Harmonic theory for flat line bundles

In this section, we summarize the theory of harmonic forms with coefficients in a flat line bundle, developed by Simpson [Simpson]. Let $A^{k}(X)$ be the space of smooth complex-valued $k$-forms on $X$. Choose a Kähler metric on $X$, with Kähler form $\omega\in A^{2}(X)$, and denote by

the associated Lefschetz operator. The metric gives rise to the $\ast$-operator

defines a Hermitian inner product on the space $A^{k}(X)$. With respect to these inner products, the adjoint of $L\colon A^{k}(X)\to A^{k+2}(X)$ is the operator $\Lambda\colon A^{k}(X)\to A^{k-2}(X)$. Likewise, the adjoint of the exterior derivative $d\colon A^{k}(X)\to A^{k+1}(X)$ is the operator $d^{\ast}\colon A^{k}(X)\to A^{k-1}(X)$, described more explicitly as $d^{\ast}\alpha=-\ast d\ast\alpha$. We use the notation $d=\partial+\bar{\partial}$ for the decomposition of $d$ by type; thus $\partial$ maps $(p,q)$-forms to $(p+1,q)$-forms, and $\bar{\partial}$ maps $(p,q)$-forms to $(p,q+1)$-forms.

and both the $\ast$-operator and the inner product induced by the harmonic metric agree with the standard ones defined above.

As before, we let $d_{\tau}=d+\tau$ be the operator encoding the complex structure and integrable connection on $(L,\nabla)$; concretely,

Let $d^{\ast}_{\tau}\colon A^{k}(X)\to A^{k-1}(X)$ be the adjoint of $d_{\tau}$ with respect to the inner products, and let $\Delta_{\tau}=d_{\tau}d^{\ast}_{\tau}+d^{\ast}_{\tau}d_{\tau}$ be the Laplace operator; it is an elliptic operator of second order. If we denote by

the space of $d_{\tau}$-harmonic $k$-forms, then $\mathcal{H}_{\tau}^{k}$ is finite-dimensional, and Hodge theory gives us a decomposition

orthogonal with respect to the inner product on $A^{k}(X)$. Let $H_{\tau}\colon A^{k}(X)\to\mathcal{H}_{\tau}^{k}$ be the orthogonal projection to the space of harmonic forms. It is not hard to see that any $\alpha\in A^{k}(X)$ can be uniquely written in the form

where $G_{\tau}\alpha\in\Delta_{\tau}A^{k}(X)$ is the so-called Green’s operator. The uniqueness of the decomposition implies that $d_{\tau}G_{\tau}=G_{\tau}d_{\tau}$ and $d^{\ast}_{\tau}G_{\tau}=G_{\tau}d^{\ast}_{\tau}$.

Following Simpson, we have a decomposition $d_{\tau}=\partial_{\tau}+\bar{\partial}_{\tau}$, where

The justification for defining these two peculiar operators is that they satisfy the usual Kähler identities (which fail for the naive choice $\partial+\tau^{1,0}$ and $\bar{\partial}+\tau^{0,1}$).

We have $\partial_{\tau}\partial_{\tau}=\bar{\partial}_{\tau}\bar{\partial}_{\tau}=\partial_{\tau}\bar{\partial}_{\tau}+\bar{\partial}_{\tau}\partial_{\tau}=0$.

Let $\partial^{\ast}_{\tau}$ and $\bar{\partial}^{\ast}_{\tau}$ denote the adjoints of $\partial_{\tau}$ and $\bar{\partial}_{\tau}$, respectively. Then the first-order Kähler identities

We have $\bar{\partial}_{\tau}\partial^{\ast}_{\tau}+\partial^{\ast}_{\tau}\bar{\partial}_{\tau}=\partial_{\tau}\bar{\partial}^{\ast}_{\tau}+\bar{\partial}^{\ast}_{\tau}\partial_{\tau}=0$.

and consequently, $d_{\tau}$-harmonic forms are both $\partial_{\tau}$-closed and $\partial^{\ast}_{\tau}$-closed.

We have $H\partial_{\tau}=H\bar{\partial}_{\tau}=H\partial^{\ast}_{\tau}=H\bar{\partial}^{\ast}_{\tau}=0$.

The Green’s operator $G_{\tau}$ commutes with $\partial_{\tau}$, $\bar{\partial}_{\tau}$, $\partial^{\ast}_{\tau}$, and $\bar{\partial}^{\ast}_{\tau}$.

Note that $\theta$ is holomorphic on account of $\bar{\partial}_{\tau}\bar{\partial}_{\tau}=0$. The complex structure on the original flat line bundle is defined by the operator $\bar{\partial}+\tau^{0,1}$, which means that the two line bundles are different unless $\tau^{0,1}=-\overline{\tau^{1,0}}$.

Harmonic theory can be used to solve equations involving $\partial_{\tau}$ (or any of the other operators), as follows. Suppose that we are given an equation of the form $\partial_{\tau}\alpha=\beta$. A necessary and sufficient condition for the existence of a solution $\alpha$ is that $\partial_{\tau}\beta=0$ and $H_{\tau}\beta=0$. If this is the case, then among all possible solutions, there is a unique one that is $\partial^{\ast}_{\tau}$-exact, namely $2\partial^{\ast}_{\tau}G_{\tau}\beta$. (In fact, this is the solution of minimal norm.) Note that we can always define $\alpha=2\partial^{\ast}_{\tau}G_{\tau}\beta$; but since

we only obtain a solution to the original equation when $H_{\tau}\beta=0$ and $\partial_{\tau}\beta=0$. This idea will appear again in the construction below.

Sobolev spaces and norm estimates

At some point of the construction below, we will need to prove the convergence of certain power series. This requires estimates for the norms of the two operators $\bar{\partial}_{\tau}$ and $G_{\tau}$ introduced above, which hold in suitable Sobolev spaces. Since this is standard material in the theory of partial differential equations, we only give the briefest possible summary; all the results that we use can be found, for example, in [Wells]*Chapter IV.

From the Kähler metric on $X$, we get an $L^{2}$-norm on the space $A^{k}(X)$ of smooth $k$-forms, by the formula

It is equivalent to the usual $L^{2}$-norm, defined using partitions of unity. There is also a whole family of higher Sobolev norms: for $\alpha\in A^{k}(X)$, the $m$-th order Sobolev norm $\lVert\alpha\rVert_{m}$ controls the $L^{2}$-norms of all derivatives of $\alpha$ of order at most $m$. The Sobolev space $W_{m}^{k}(X)$ is the completion of $A^{k}(X)$ with respect to the norm $\lVert-\rVert_{m}$; it is a Hilbert space. Elements of $W_{m}^{k}(X)$ may be viewed as $k$-forms $\alpha$ with measurable coefficients, all of whose weak derivatives of order at most $m$ are square-integrable. Here is the first result from analysis that we need.