arXiv:1112.3058v1 [math.AG] 13 Dec 2011

GENERIC VANISHING THEORY VIA MIXED HODGE
MODULESMIHNEA POPA AND CHRISTIAN SCHNELL

Abstract. We extend the dimension and strong linearity results of generic
vanishing theory to bundles of holomorphic forms and rank one local systems,
and more generally to certain coherent sheaves of Hodge-theoretic origin associated to irregular varieties. Our main tools are Saito’s mixed Hodge modules,
the Fourier-Mukai transform for D-modules on abelian varieties introduced by
Laumon and Rothstein, and Simpson’s harmonic theory for flat bundles. In
the process, we discover two natural categories of perverse coherent sheaves
on the Picard variety and on the moduli space for Higgs line bundles.

A. Introduction
1. Generic vanishing theory. The attempt to understand cohomology vanishingstatements 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 thegeneric vanishing theorem of Green and Lazarsfeld [GL87], which in a weak form
states that on a smooth complex projective variety X, the cohomology of a genericline bundle L ∈ Pic0 (X) vanishes in degrees less than dim a(X), where a : X →Alb(X) denotes the Albanese mapping of X. This theorem and its variants havefound a surprising number of applications, ranging from results about singularitiesof theta divisors [EL97] to recent work on the birational geometry of irregularvarieties, including a proof of Ueno’s conjecture [CH11].
One can consider the set of those line bundles for which the cohomology in agiven degree does not vanish, and thanks to the work of many people, the structureof these sets is very well understood. This is more precisely the content of generic
vanishing theory. Denoting, for any coherent sheaf F on X, by
V i (F ) := {L ∈ Pic0 (X) | H i (X, F ⊗ L) 6= 0} ⊆ Pic0 (X)the i-th cohomological support locus of F , its main statements are roughly thefollowing:
Dimension (D): One has codim V i (ωX ) ≥ i − dim X + dim a(X) for all i [GL87,
GL91]. This implies the generic vanishing theorem via Serre duality.
Linearity (L): The irreducible components of each V i (ωX ) are torsion translatesof abelian subvarieties of Pic0 (X) [GL91, Ara92, Sim93].
Strong linearity (SL): If p2 : X × Pic0 (X) → Pic0 (X) is the second projection,
and P is a Poincaré line bundle on X × Pic0 (X), then Rp2 ∗ P is locallyaround each point quasi-isomorphic to a linear complex [GL91]. A precise
Date: December 15, 2011.
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M. POPA AND C. SCHNELL

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 Pic0 (X) by Char(X), the algebraic group parametrizing rank one local
systems [Ara92, Sim92, Sim93]. New approaches and extensions for the theory on
Pic0 (X) have been introduced more recently, for example in [CH02, Hac04, PP11].
On the other hand, important gaps have remained in our understanding of someof the most basic objects. For instance, while (L) is also known for the sheafof holomorphic p-forms ΩpX with p < n, a good generic Nakano-type vanishingstatement as in (D) has eluded previous efforts, despite several partial results [GL87,
PP11]. 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 D-modules associatedto mixed Hodge modules on abelian varieties. In fact, there is a version of theFourier-Mukai transform for D-modules, introduced by Laumon [Lau96] and Rothstein [Rot96]; it takes D-modules on an abelian variety to complexes of coherent
sheaves on A♯ , 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 D-module OX on an irregular variety X.
2. 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 discovered by Hacon [Hac04]. It goes as follows.
Let A = Alb(X) denote the Albanese variety of an irregular smooth complex
projective variety X, and a : X → A its Albanese mapping (for some choice ofb = Pic0 (A) denote the dual abelianbase point, which does not matter here). Let A
variety. Using a well-known theorem of Kollár on the splitting of the direct imageRa∗ ωX in Dbcoh (OA ) and standard manipulations, it is enough to prove that
codim V ℓ (Ri a∗ ωX ) ≥ ℓfor all i = 0, 1, . . . , k := dim X − dim a(X).

In terms of the Fourier-Mukai transform RΦP : Dbcoh (OA ) → Dbcoh (OAb), this isequivalent, via base change arguments, to the statement that
codim Supp Rℓ ΦP (Ri a∗ ωX ) ≥ ℓin the same range. Now the sheaves Ri a∗ ωX still satisfy a Kodaira-type vanishingtheorem, and together with the special geometry of abelian varieties this implies
after some work that the Fourier-Mukai transform of Ri a∗ ωX is the dual of ab i.e.
coherent sheaf Fi on A,
RΦP (Ri a∗ ωX ) ≃ RHom(Fi , OAb).

The desired inequality for the codimension of the support becomes
codim Supp Rℓ ΦP (Ri a∗ ωX ) = codim Supp Ext ℓ (Fi , OAb) ≥ ℓ,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.

GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

3

One of the subjects of this paper is to use this general framework in order toprove a generic vanishing theorem for general objects of Hodge-theoretic origin,and in particular for the sheaves ΩpX with p < dim X. The role of Kollár’s theorem is played by the decomposition theorem [BBD82], or more precisely by its
Hodge-theoretic version due to Morihiko Saito [Sai90]. This is one main reason
why mixed Hodge modules form a natural setting here. Another is the existenceof a very general Kodaira-type vanishing theorem for mixed Hodge modules, again
due to Saito, which becomes particularly useful on abelian varieties. This vanishingtheorem 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 ΩpX , one needs a result by Laumon andSaito on the behavior of filtered D-modules under direct images, which only works
well in the case of D-modules that underlie mixed Hodge modules.
3. The main results. Let us now give a summary of the results we obtain. Thereare essentially two parts: vanishing and dimension results, for which which Hodge
modules are crucially needed, and linearity results, which apply to certain Hodge
modules, but for which the general theory of D-modules and the harmonic theoryof flat line bundles suffice in the proofs. The theory of mixed Hodge modules isreviewed in §4 below.
The starting point is a general Kodaira-type vanishing theorem for the gradedpieces 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 grF
k M, where (M, F ) is any filtered D-module underlying
a mixed Hodge module on A (see Lemma 8.1 below). We use this observation
to produce natural classes of perverse coherent sheaves [AB10, Kas04] on the dual0b and on the parameter space for Higgs line bundles A×Hbabelian variety A,(A, Ω1A ).
We first show that every mixed Hodge module on A gives rise to a collection ofb (with respect to the dual standard t-structure).
perverse coherent sheaves on A
Theorem 3.1. Let A be a complex abelian variety, and M a mixed Hodge module
on A with underlying filtered D-module (M, F ). Then for each k ∈ Z, the coherent
sheaf grF
k M is a GV -sheaf on A, i.e.
codim V i (grF
k M) ≥ i for all i.
Consequently, its Fourier-Mukai transform RΦP (grF
k M) is a perverse coherentbsheaf on A.

This uses Hacon’s general strategy, as in §2 above, and the correspondence established in [Pop09,PP11] between objects satisfying generic vanishing (or GV-objects)
and perverse coherent sheaves in the sense of Kashiwara [Kas04].
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 [Sai88], extending the well-known result of [BBD82], and a formuladue to Laumon [Lau85] for the behavior of the associated graded objects under
projective direct images (which is true for mixed Hodge modules).

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M. POPA AND C. SCHNELL

Our main results in this direction are the following. Let X be a smooth complex
projective variety of dimension n, with nonzero irregularity g = h1 (X, OX ). Let
a : X → A = Alb(X) be the Albanese map of X. Consider the defect of semismallness of the Albanese map a : X → A, which is defined by the formulaδ(a) = max 2ℓ − dim X + dim Aℓ ,


ℓ∈N

−1

where Aℓ = y ∈ A dim f (y) ≥ ℓ for ℓ ∈ N. By applying the decomposition
theorem and our results about mixed Hodge modules on abelian varieties to thedirect image of the trivial Hodge module QH
X [n], we obtain the following theorem.
Theorem 3.2. Let X be a smooth complex projective variety of dimension n. Then
codim V q (ΩpX ) ≥ |p + q − n| − δ(a).
for every p, q ∈ N.
A slightly stronger (and optimal) result, involving a natural invariant k(X) ≤δ(a) coming from the direct sum decomposition of a∗ QH
X [n], is given in Theorem
10.2. The statement above is the appropriate generalization of the original generic
vanishing theorem [GL87], which dealt with the case p = n. Note that, unlike in[GL87], the codimension bound depends on the entire Albanese mapping, not juston the dimension of the generic fiber. In the language of [PP11], our theorem is
equivalent to the fact that, for each p ∈ Z, the bundle ΩpX is a GVp−n−δ(a) -sheafwith respect to the Fourier-Mukai transform induced by the Poincaré bundle on
X × Pic0 (X). Since the condition δ(a) = 0 is equivalent to the Albanese map being
semi-small, in particular one obtains:
Corollary 3.3. Suppose that the Albanese map of X is semi-small. Then
codim V q (ΩpX ) ≥ |p + q − n|,for every p, q ∈ N, and so X satisfies the generic Nakano vanishing theorem.
Unlike in the case of ωX , it is not sufficient to assume that the Albanese map isgenerically finite over its image; this was already pointed out in [GL87]. Nevertheless, our method also recovers the stronger statement for ωX [GL87] and its higher
direct images [Hac04] (see the end of §10). This is due to the special properties ofthe first non-zero piece of the Hodge filtration on mixed Hodge modules, establishedby Saito.
This approach also leads to a dimension theorem of type (D) for rank one local
systems. Let Char(X) = Hom(π1 (X), C∗ ) be the algebraic group of characters ofX, and for each i consider the cohomological support lociΣk (X) = ρ ∈ Char(X) H k (X, Cρ ) 6= 0 ,where Cρ denotes the local system of rank one associated to a character ρ.

Theorem 3.4. Let X be a smooth complex projective variety. Then for each k ∈ N,
codimChar(X) Σk (X) ≥ 2 |k − dim X| − k(X) .

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, provedby Simpson [Sim92, Sim93] and Arapura [Ara92]; see §12.

GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

5

Note. While editing this paper, we learned of the very interesting preprint [KW11]
by T. Krämer and R. Weissauer, who prove vanishing theorems for perverse sheaves
on abelian varieties. They also obtain a generic vanishing theorem for ΩpX and forrank one local systems, involving the same quantity δ(a) as in Theorem 3.2, but
without precise codimension bounds for the cohomological support loci. Theirmethods are very different from ours.
Two additional theorems complete the picture, by describing in detail the FourierMukai transform of the trivial D-module OX ; they include results of type (D), (L)and (SL) on the space of rank one local systems on X. Here it is important to consider two different kinds of Fourier-Mukai transforms, corresponding in Simpson’s
terminology [Sim93] to the Dolbeault realization (via Higgs bundles) and the de
Rham realization (via line bundles with integrable connection) of Char(X).
Setting V = H 0 (A, Ω1A ), one can naturally extend the usual Fourier-Mukai functor Dbcoh (OA ) → Dbcoh (OAb) to a relative transform (see §9).
RΦP : Dbcoh OA×V → Dbcoh OA×Vb
One of the two theorems describes how the complex of filtered D-modules a∗ (OX , F )bunderlying a∗ QH
X [n] ∈ D MHM(A) behaves with respect to this transform. We firstshow in Proposition 9.1 below that the associated graded complex satisfiesih•−g→ Ω1X ⊗ S •−g+1 → · · · → ΩnX ⊗ S •−g+n ,grF• a∗ (OX , F ) ≃ Ra∗ OX ⊗ S

where S = Sym V ∗ , and the complex in brackets is placed in degrees −n, . . . , 0, withdifferential induced by the evaluation morphism OX ⊗ V → Ω1X . Since this is acomplex of finitely generated graded modules over Sym ΘA , it naturally corresponds
to a complex of coherent sheaves on cotangent bundle T ∗ A = A × V , namelyhi
C = R(a × id)∗ p∗1 OX → p∗1 Ω1X → · · · → p∗1 ΩnX .
Theorem 3.5. Let a : X → A be the Albanese map of a smooth complex projective
variety of dimension n, and let p1 : X × V → X be the first projection.
(i) In the derived category Dbcoh (OA×V ), we have a non-canonical splitting
M
Ci,j [−i]C ≃i,jwhere each Ci,j is a Cohen-Macaulay sheaf of dimension dim A.
(ii) The support of each RΦP Ci,j is a finite union of torsion translates of tripleb × V , subject to the inequalities
tori in A
codim Supp Rℓ ΦP Ci,j ≥ 2ℓ(iii) The dual objects RHom RΦP Ci,j , OA×Vb

for all ℓ ∈ Z.

also satisfy (ii).

A more refined version of this can be found in Theorem 13.2 below, where wealso interpret the result in terms of perverse coherent sheaves: the objects RΦP Ci,jbelong to a self-dual subcategory of the category of perverse coherent sheaves, withrespect to a certain t-structure on Dbcoh (OA×V ) introduced in §6. Note that thist-structure is different from the “dual standard t-structure” that appears in thegeneric vanishing theory of topologically trivial line bundles [Pop09].

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M. POPA AND C. SCHNELL

The second of the two theorems is best stated in terms of the generalized FourierMukai transform for D-modules on abelian varieties, introduced by Laumon [Lau96]
and Rothstein [Rot96]. Their work gives an equivalence of categoriesRΦP ♯ : Dbcoh (DA ) → Dbcoh (OA♯ ),where A♯ is the moduli space of line bundles with integrable connection on A, with
universal cover W = H 1 (A, C) = H 1 (X, C). By composing with the Albanese
mapping, there is also an induced functorRΦP ♯ : Dbcoh (DX ) → Dbcoh (OA♯ ).
We review the construction, both algebraically and analytically, in §15 below. This
Fourier-Mukai transform is the right context for a strong linearity result (SL) for the
D-module OX , extending the result for topologically trivial line bundles in [GL91].
Theorem 3.6. Let X be a smooth complex projective variety. Then the generalized
Fourier-Mukai transform RΦP ♯ OX ∈ Dbcoh (OA♯ ) is locally, in the analytic topology,
quasi-isomorphic to a linear complex. More precisely, let (L, ∇) be any line bundle
with integrable connection on X, and L the associated local system. Then the
pullback of RΦP ♯ OX from a neighborhood of (L, ∇) to the covering space W is
quasi-isomorphic to (restriction of ) the complex
H 0 (X, L) ⊗ OW → H 1 (X, L) ⊗ OW → · · · → H 2n (X, L) ⊗ OW ,with differential given by the formulak

H (X, L) ⊗ OW → H

k+1(X, L) ⊗ OW ,α ⊗ f 7→2g
X

(εj ∧ α) ⊗ zj f.

j=1

Here ε1 , . . . , ε2g is any basis of H 1 (X, C), and z1 , . . . , z2g are the corresponding
holomorphic coordinates on the affine space W .
As discussed in §20, 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 RΦP ♯ OX coming from the decomposition
M
Mi,j [−i] ∈ Db MHM(A)a∗ Q HX [n] ≃ihave the same linearity property (see Corollary 15.3). Note also that using basechange for local systems and the description of tangent cones to cohomologicalsupport loci as in [Lib02], via arguments as in[GL91] §4 (which we will not repeathere), Theorem 3.6 gives another proof of the linearity to the cohomological support
loci Σk (X), i.e. the statement of type (L).
Our proof of Theorem 3.6 relies on the harmonic theory for flat line bundles
developed by Simpson [Sim92]. As in [GL91], the idea is that after pulling thecomplex RΦP ♯ OX back to the universal covering space of A♯ , one can use harmonic
forms to construct a linear complex that is quasi-isomorphic to the pullback ina neighborhood of a given point. Additional technical difficulties arise howeverfrom 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 [GL91], andlet to a quasi-isomorphism naturally being present in the picture). The new insightin this part of the paper is that a quasi-isomorphism can still be constructed bymore involved analytic methods.

GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

7

The last part of the paper contains a few statements extending our results tomore general classes of 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 thank Takuro Mochizuki and Keiji Oguiso for usefulconversations.
B. Preliminaries
4. Mixed Hodge modules. In this section, we recall a few aspects of Morihiko
Saito’s theory of mixed Hodge modules [Sai90] 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 [Sai94]. We only discuss the case of a complex algebraic variety
X, say of dimension n.
Saito defines an abelian category MHM(X) of graded-polarizable mixed Hodge
modules on X. A mixed Hodge module is, roughly speaking, a variation of mixed
Hodge structure with singularities. Over a one-point space it is the same thing asa graded-polarizable mixed Hodge structure with coefficients in Q.
As in classical Hodge theory, there are pure and mixed objects. We shall denoteby HMℓ (X) the subcategory of polarizable (pure) Hodge modules of weight ℓ. On
a one-point space, such an object is just a polarizable Hodge structure of weightℓ. More generally, any polarizable variation of Hodge structure of weight k on a
Zariski-open subset of some irreducible subvariety Z ⊆ X uniquely extends to a
Hodge module of weight dim Z + k on X. The existence of polarizations makes thecategory HMℓ (X) semi-simple: each object admits a decomposition by support, andsimple objects with support equal to an irreducible subvariety Z ⊆ X are obtained
from polarizable variations of Hodge structure on Zariski-open subsets of Z.
When M ∈ HMℓ (X) is pure of weight ℓ, and we are in the presence of a projective
morphism f : X → Y , Saito shows that the direct image f∗ M ∈ Db MHM(Y ) splits
non-canonically into the sum of its cohomology objects H i f∗ M ∈ HMℓ+i (Y ). The
resulting isomorphismM
H i f∗ M [−i]f∗ M ≃iis the analogue for Hodge modules of the decomposition theorem of [BBD82]. The
proof of this result with methods from algebraic analysis and D-modules is one of
the main achievements of Saito’s theory.
The precise definition of a mixed Hodge module is very involved; it uses regularholonomic D-modules, perverse sheaves, and the theory of nearby and vanishingcycles. The familiar equivalence between local systems and flat vector bundles
is replaced by the Riemann-Hilbert correspondence between perverse sheaves andregular holonomic D-modules. Let M be a D-module on X; in this paper, thisalways means a left module over the sheaf of algebraic differential operators DX .
Recall that the de Rham complex of M is the C-linear complexhi
DRX (M) = M → Ω1X ⊗ M → · · · → ΩnX ⊗ M ,placed in degrees −n, . . . , 0. When M is holonomic, DRX (M) is constructible
and satisfies the axioms for a perverse sheaf with coefficients in C. According to8

M. POPA AND C. SCHNELL

the Riemann-Hilbert correspondence of Kashiwara and Mebkhout, the categoryof regular holonomic D-modules is equivalent to the category of perverse sheaves
Perv(CX ) through the functor M 7→ DRX (M). Internally, a mixed Hodge module
M on a smooth complex algebraic variety X has three components:(1) A regular holonomic DX -module M, together with a good filtration F• M
F
by OX -coherent subsheaves such that grF• M is coherent over gr• DX . Thisfiltration plays the role of a Hodge filtration on M.
(2) A perverse sheaf K ∈ Perv(QX ), together with an isomorphism
α : DRX (M) → K ⊗Q C.
This isomorphism plays the role of a Q-structure on the D-module M.
(When M is a mixed Hodge structure, K is the underlying Q-vector space;when M corresponds to a polarizable variation of Hodge structure H on
a Zariski-open subset of Z ⊆ X, K is the intersection complex ICZ (H) of
the underlying local system.)(3) A finite increasing weight filtration W• M of M by objects of the same kind,compatible with α, such that the graded quotients grWℓ M = Wℓ M/Wℓ−1 M
belong to HMℓ (X).
These components are subject to several conditions, which are defined by induction on the dimension of the support of 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 M with respect to any locallydefined holomorphic function are again mixed Hodge modules (now on a variety
of dimension n − 1); the existence of polarizations; etc. An important fact is thatthese local conditions are preserved after taking direct images, and that they are
sufficient to obtain global results such as the decomposition theorem.
The most basic Hodge module on a smooth variety X is the trivial Hodge module,
denoted by QH
X [n]. Its underlying D-module is the structure sheaf OX , with thetrivial filtration F0 OX = OX ; its underlying perverse sheaf is QX [n], the constantlocal system placed in degree −n. The comparison isomorphism is nothing but thewell-known fact that the usual holomorphic de Rham complex
OX → Ω1X → Ω2X → · · · → ΩnX
is a resolution of the constant local system CX . In general, all admissible variationsof mixed Hodge structures are mixed Hodge modules, and any mixed Hodge module
is generically one such. The usual cohomology groups of X can be obtained as thecohomology of the complex p∗ p∗ QH ∈ Db MHM(pt ), where p : X → pt denotes themap to a one-point space, and QH is the Hodge structure of weight zero on Q.
There are similar expressions for cohomology with compact supports, intersection
cohomology, and so forth, leading to a uniform description of the mixed Hodge
structures on all of these groups.
5. Three theorems about mixed Hodge modules. In this section, we recall
from the literature three useful theorems about the associated graded object grF• M,
for a filtered D-module (M, F ) underlying a mixed Hodge module.
During the discussion, X will be a smooth complex projective variety of dimension n, and M ∈ MHM(X) a mixed Hodge module on X. As usual, we denote theunderlying filtered D-module by (M, F ). Recall that the associated graded of theGENERIC VANISHING THEORY VIA MIXED HODGE MODULES

9sheaf of differential operators DX , with respect to the filtration by order of differential operators, is isomorphic to A•X = Sym• ΘX , the symmetric algebra of thetangent sheaf of X. Since grF• M is finitely generated over this sheaf of algebras,
it defines a coherent sheaf C (M, F ) on the cotangent bundle T ∗ X. The supportof this sheaf is the characteristic variety of the D-module M, and therefore of pure
dimension n because M is holonomic.
The first of the three theorems is Saito’s generalization of the Kodaira vanishingtheorem. Before we state it, observe that the filtration F• M is compatible with
the D-module structure on M, and therefore induces a filtration on the de Rhamcomplex of M by the formula
h
i
(5.1)
Fk DRX (M) = Fk M → Ω1X ⊗ Fk+1 M → · · · → ΩnX ⊗ Fk+n M .
The associated graded complex for the filtration in (5.1) is
i
h
n
F
F
F
1
grF
k DRX (M) = grk M → ΩX ⊗ grk+1 M → · · · → ΩX ⊗ grk+n M ,which is now a complex of coherent sheaves of OX -modules in degrees −n, . . . , 0.
This complex satisfies the following Kodaira-type vanishing theorem.
Theorem 5.2 (Saito). Let (M, F ) be the filtered D-module underlying a mixed
Hodge module on a smooth projective variety X, and let L be any ample line bundle.
(1) One has Hi X, grF
k DRX (M) ⊗ L = 0 for all i > 0.
−1(2) One has Hi X, grF
DR
(M)
⊗
L
= 0 for all i < 0.
X
kProof. The proof works by reducing the assertion to a vanishing theorem for perverse sheaves on affine varieties, with the help of Saito’s formalism. More details
can be found in [Sai90, Proposition 2.33].

Example 5.3. For the trivial Hodge module M = QH
X [n] on X, we have M = OX ,and therefore grF−n DRX (OX ) = ωX . This shows that Theorem 5.2 generalizes thep
Kodaira vanishing theorem. Since grF−p DRX (OX ) = ΩX [n − p], it also generalizes
the Nakano vanishing theorem.
The second theorem gives more information about the coherent sheaf C (M, F )on the cotangent bundle of X. Before stating it, we recall the definition of the
Verdier dual M ′ = DX M of a mixed Hodge module. If rat M is the perverse sheaf
underlying M , then rat M ′ is simply the usual topological Verdier dual. On thelevel of D-modules, note that since M is left D-module, the dual complex

RHom DX M, DX [n]is naturally a complex of right D-modules; since M is holonomic, it is quasiisomorphic to a single right D-module. If M′ denotes the left D-module underlying
the Verdier dual M ′ , then that right D-module is ωX ⊗OX M′ . Thus we have
RHom DX M, DX [n] ≃ ωX ⊗OX M′ ,where the right D-module structure on ωX ⊗ M′ is given by the rule ξ · (ω ⊗ m′ ) =(ξω)⊗m′ −ω ⊗(ξm′ ) for ξ ∈ ΘX . The Hodge filtration on M induces a filtration on
the dual complex and hence on M′ . The following result shows that this induced
filtration is well-behaved, in a way that makes duality and passage to the associatedgraded compatible with each other.

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M. POPA AND C. SCHNELL

Theorem 5.4 (Saito). Let M be a mixed Hodge module on a smooth complex
algebraic variety X of dimension n, and let M ′ denote its Verdier dual. ThenF′•
RHom A•X grF• M, AX [n] ≃ ωX ⊗OX gr•+2n M ,where sections of Symk ΘX act with an extra factor of (−1)k on the right-hand side.
If we consider both sides as coherent sheaves on T ∗ X, we obtain

RHom C (M, F ), OT ∗ X [n] ≃ p∗ ωX ⊗ (−1)∗T ∗ X C (M′ , F ),where p : T ∗ X → X is the projection. In particular, C (M, F ) is always a CohenMacaulay sheaf of dimension n.
Proof. That the coherent sheaf C (M, F ) on T ∗ X is Cohen-Macaulay is proved in[Sai88, Lemme 5.1.13]. For an explanation of how this fact implies the formula forthe dual of the graded module grF
• M, see [Sch11, §3.1].
The last of the three theorems gives a formula for the associated graded objectof f∗ M , where f : X → Y is a projective morphism between two smooth complex
algebraic varieties. The formula itself first appears in a paper by Laumon [Lau85,
Construction 2.3.2], but it is not true in the generality claimed there (that is to say,for arbitrary filtered D-modules).
Before stating the precise result, we give an informal version. The following
diagram of morphisms, induced by f , will be used throughout:df

T ∗X ✛T ∗ Y ×Y X
p1

p2

✲ X
f

❄
❄
p
✲ Y
T Y
Let M ∈ MHM(X) be a mixed Hodge module, let (M, F ) be its underlying filtered
D-module, and write C (M, F ) for the associated coherent sheaf on T ∗ X. We shalldenote the complex of filtered holonomic D-modules underlying the direct imagef∗ M ∈ Db MHM(Y ) by the symbol f∗ (M, F ). Then Laumon’s formula claims thatC f∗ (M, F ) ≃ Rp1∗ Ldf ∗ C (M, F ) ⊗ p∗2 ωX/Y∗

as objects in the derived category Dbcoh (OT ∗ Y ). Unfortunately, the passage to coherent sheaves on the cotangent bundle loses the information about the grading;
it is therefore better to work directly with the graded modules. As above, wewrite p : T ∗ X → X for the projection, and denote the graded sheaf of OX -algebrasp∗ OT ∗ X = Sym ΘX by the symbol AX .
Theorem 5.5 (Laumon). Let f : X → Y be a projective morphism between smoothcomplex algebraic varieties, and let M ∈ MHM(X). Then with notation as above,


∗
F
.
f
A
gr
M
⊗
ω
⊗
grF
Rf
(M,
F
)
≃
Rf
Y
A
O
∗
∗
X/Y
X
X
•+dim X−dim Y•
Proof. This can be easily proved using Saito’s formalism of induced D-modules;
both the factor of ωX/Y and the shift in the grading come from the transformation
between left and right 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 embeddingf : X ֒→ Y ; (2) for a projection f : Y × Z → Y with Z smooth and projective.

GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

11

We first consider the case where f : X ֒→ Y is a regular closed embedding ofcodimension r. Here ωX/Y ≃ det NX|Y . Working locally, we may assume without
loss of generality that Y = X × Cr . Let t1 , . . . , tr denote the coordinates on Cr ,and let ∂1 , . . . , ∂r be the corresponding vector fields. Thenf ∗ AY = AX [∂1 , . . . , ∂r ],and so the formula we need to prove is that


F
grF
• f∗ (M, F ) ≃ f∗ gr•−r M ⊗AX AX [∂1 , . . . , ∂r ] .
By [Sai88, p. 850], we have f∗ (M, F ) ≃ M[∂1 , . . . , ∂r ], with filtration given byXf∗ Fk M ⊗ ∂ ν .
Fp f∗ (M, F ) ≃k+|ν|≤p−r

Consequently, we get
grF
p f∗ (M, F ) ≃

M

ν∈Nr


ν
f∗ grFp−r−|ν| M ⊗ ∂ ,which is the desired formula. Globally, the factor of det NX|Y is needed to makethe above isomorphism coordinate independent.
Next, consider the case where X = Y × Z, with Z smooth and projective ofdimension r, and f = p1 . Then ωX/Y ≃ p∗2 ωZ . In this case, we havef∗ (M, F ) = Rp1∗ DRY ×Z/Y (M),where DRY ×Z/Y (M) is the relative de Rham complexih
M → Ω1Y ×Z/Y ⊗ M → · · · → ΩrY ×Z/Y ⊗ M ,supported in degrees −r, . . . , 0. As in (5.1), the Hodge filtration on the D-module
M induces a filtration on the relative de Rham complex by the formulaih
Fk DRY ×Z/Y (M) = Fk M → Ω1Y ×Z/Y ⊗ Fk+1 M → · · · → ΩrY ×Z/Y ⊗ Fk+r M .

Now the key point is that since Z is smooth and projective, the induced filtrationon the direct image complex is strict by [Sai90, Theorem 2.14]. It follows that
h
i
F
∗ 1
F
∗ r
F
gr
M
→
p
Ω
⊗
gr
M
→
·
·
·
→
p
Ω
⊗
gr
M
.
grF
f
(M,
F
)
≃
Rp
∗
1∗
•
2 Z
•+1
2 Z
•+r
•

On the other hand, a graded locally free resolution of ωZ as a graded AZ -module
is given by the complex
i
h
•−r
•−r+1
AZ
→ Ω1Z ⊗ AZ
→ · · · → ΩrZ ⊗ A•Z ,again in degrees −r, . . . , 0. Therefore ωX/Y ⊗OX f ∗ AY is naturally resolved, as a
graded AX -module, by the complex
i
h
•−r+1
∗ 1
∗ r
•
,
A•−r
→
p
Ω
⊗
A
→
·
·
·
→
p
Ω
⊗
A
2
Z
2
Z
X
X
X
∗
and so grF•+r M ⊗AX f AY ⊗OX ωX/Y is represented by the complex
h
i
∗ 1
F
∗ r
F
grF
• M → p2 ΩZ ⊗ gr•+1 M → · · · → p2 ΩZ ⊗ gr•+r M ,from which the desired formula follows immediately.

12

M. POPA AND C. SCHNELL

6. Perverse coherent sheaves. Let X be a smooth complex algebraic variety. In
this section, we record some information about perverse t-structures on Dbcoh (OX ),emphasizing two that are of interest to us here. We follow the notation introduced
by Kashiwara [Kas04]. For a (possibly non-closed) point x of the scheme X, wewrite κ(x) for the residue field at the point, ix : Spec κ(x) ֒→ X for the inclusion,and codim(x) = dim OX,x for the codimension of the closed subvariety {x}.
A supporting function on X is a function p : X → Z from the topological space ofthe scheme X to the integers, with the property that p(y) ≥ p(x) whenever y ∈ {x}.
Given such a supporting function, one defines two families of subcategories≤k+p(x)p ≤k
κ(x) for all x ∈ X
Dcoh (OX ) = M ∈ Dbcoh (OX ) Li∗x M ∈ Dcoh
and

p

D≥k
coh (OX ) =M ∈ Dbcoh (OX )≥k+p(x)

Ri!x M ∈ Dcohκ(x) for all x ∈ X .

The following fundamental result is proved in [Kas04, Theorem 5.9] and, based on
an idea of Deligne, in [AB10, Theorem 3.10].
Theorem 6.1. The above subcategories define a bounded t-structure on Dbcoh (OX )iff the supporting function has the property that p(y) − p(x) ≤ codim(y) − codim(x)for every pair of points x, y ∈ X with y ∈ {x}.
For example, p = 0 corresponds to the standard t-structure on Dbcoh (OX ). An
equivalent way of putting the condition in Theorem 6.1 is that the dual functionp̂(x) = codim(x) − p(x) should again be a supporting function. If that is the case,
one has the identities


p̂ ≤k
Dcoh (OX ) = RHom p D≥−kcoh (OX ), OX


p̂ ≥k
(O
),
O
Dcoh (OX ) = RHom p D≤−k
,
X
X
coh

which means that the duality functor RHom(−, OX ) exchanges the two perverse
t-structures defined by p and p̂. The heart of the t-structure defined by p is denotedpp ≥0
Coh(OX ) = p D≤0coh (OX ) ∩ Dcoh (OX ),and is called the abelian category of p-perverse coherent sheaves.
We are interested in two special cases of Kashiwara’s result. One is that theset of objects E ∈ Dbcoh (OX ) with codim Supp H i (E) ≥ i for all i forms part of at-structure (the “dual standard t-structure” in Kashiwara’s terminology); the otheris that the same is true for the set of objects with codim Supp H i (E) ≥ 2i for all i.
This can be formalized in the following way.
We first define a supporting function c : X → Z by the formula c(x) = codim(x).
Since the dual function is ĉ = 0, it is clear that c defines a t-structure on Dbcoh (OX ),namely the dual of the standard t-structure. The following result by Kashiwara[Kas04, Proposition 4.3] gives an alternative description of c Coh(OX ).
Lemma 6.2. The following three conditions on E ∈ Dbcoh (OX ) are equivalent:(1) E belongs to c Coh(OX ).
(2) The dual object RHom(E, OX ) is a coherent sheaf.
(3) E is quasi-isomorphic to a bounded complex of flat OX -modules in nonnegative degrees, and satisfies codim Supp H i (E) ≥ i for every i ≥ 0.

GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

13

We now define a second function m : X → Z by the formulam(x) = 12 codim(x) .

It is easily verified that both m and the dual functionm̂(x) = 12 codim(x)are supporting functions. As a consequence of Theorem 6.1, m defines a boundedt-structure on Dbcoh (OX ); objects of the heart m Coh(OX ) will be called m-perverse
coherent sheaves. (We use this letter because m and m̂ are as close as one can get
to “middle perversity”. Since the equality m = m̂ can never be satisfied, there isof course no actual middle perversity for coherent sheaves.)
The next lemma follows easily from [Kas04, Lemma 5.5].
Lemma 6.3. The perverse t-structures defined by m and m̂ satisfym ≤kDcoh (OX ) = E ∈ Dbcoh (X) codim Supp H i (E) ≥ 2(i − k) for all i ∈ Zm̂ ≤kDcoh (OX ) = E ∈ Dbcoh (X) codim Supp H i (E) ≥ 2(i − k) − 1 for all i ∈ Z .
By duality, this also describes the subcategories with ≥ k.

Consequently, an object E ∈ Dbcoh (OX ) is an m-perverse coherent sheaf iff
codim Supp H i (E) ≥ 2i andcodim Supp Ri Hom(E, OX ) ≥ 2i − 1for every i ∈ Z. This shows one more time that the category of m-perverse coherent
sheaves is not preserved by the duality functor RHom(−, OX ).
Lemma 6.4. If E ∈ m Coh(OX ), then E ∈ D≥0coh (OX ).
Proof. This is obvious from the fact that m(x) ≥ 0.



7. Integral functors and GV-objects. Let X and Y be smooth projective
complex varieties of dimensions n and g respectively, and let P ∈ Dbcoh (OX×Y ) bean object inducing integral functors
LRΦP : Dbcoh (OX ) → Dbcoh (OY ), RΦ(·) := RpY ∗ (p∗X (−) ⊗ P )andLRΨP : Dbcoh (OY ) → Dbcoh (OX ), RΨ(·) := RpX ∗ (p∗Y (−) ⊗ P ).
Let E ∈ Dbcoh (OX ). There is a useful cohomological criterion for checking
whether the Fourier transform RΦP E is a perverse coherent sheaf on Y with respect to the t-structure defined by c from the previous subsection. Note first that bybase change, for any sufficiently positive ample line bundle L on Y , the transformRΨP (L−1 ) is supported in degree g, and Rg ΨP (L−1 ) is a locally free sheaf on A.
The following is contained in [Pop09, Theorem 3.8 and 4.1], and is partly based on
the previous [Hac04, Theorem 1.2] and [PP11, Theorem A].
Theorem 7.1. The following are equivalent:(1) RΦP E belongs to c D≤0coh (OY ).
(2) For any sufficiently ample line bundle L on Y , and every i > 0,
H i A, E ⊗ Rg ΨP (L−1 ) = 0.

14

M. POPA AND C. SCHNELL

Moreover, if E is a sheaf—or more generally a geometric GV-object, in the language of [Pop09, Definition 3.7]—satisfying the two equivalent conditions above,
then RΦP E is a perverse sheaf in c Coh(OY ).
Definition 7.2. An object E satisfying the equivalent conditions in the Theorem
is called a GV-object (with respect to P ); if it is moreover a sheaf, then it is calleda GV-sheaf.
C. Constructing perverse coherent sheaves
8. Mixed Hodge modules on abelian varieties. Let A be a complex abelianvariety of dimension g, and let M ∈ MHM(A) be a mixed Hodge module on A.
As usual, we denote the underlying filtered holonomic D-module by (M, F ). From
Theorem 5.2, we know that the associated graded pieces of the de Rham complex
DR(M, F ) satisfy an analogue of Kodaira vanishing. A key observation is thaton abelian varieties, the same vanishing theorem holds for the individual coherent
sheaves grF
k M, due to the fact that the cotangent bundle of A is trivial. As we shallsee, this implies that each grF
k M is a GV-sheaf on A, and therefore transforms tob (with respect to the dual standard t-structure).
a perverse coherent sheaf on A
Lemma 8.1. Let (M, F ) be the filtered D-module underlying a mixed Hodge module
on A, and let L be an ample line bundle. Then for each k ∈ Z, we have
H i A, grF
k M⊗L =0

for every i > 0.

Proof. Consider for each k ∈ Z the complex of coherent sheaves
h
i
g
F
1
F
F
grF
DR
(M)
=
gr
M
→
Ω
⊗
gr
M
→
·
·
·
→
Ω
⊗
gr
M
,
A
k
k
A
k+1
k+g
A

supported in degrees −g, . . . , 0. According to Theorem 5.2, this complex has theproperty that, for i > 0,
Hi A, grF
k DRA (M) ⊗ L = 0.

⊕g, one can deduce the asserted vanishing theorem
Using the fact that Ω1A ≃ OA
F
for the individual sheaves grk M by induction on k. Indeed, since grF
k M = 0 fork ≪ 0, inductively one has for each k a distinguished triangle
F
Ek → grF
k DRA (M) → grk+g M → Ek [1],with Ek an object satisfying Hi (A, Ek ⊗ L) = 0.



We now obtain the first theorem of the introduction, by combining Lemma 8.1
with Theorem 7.1 and a trick invented by Mukai. As mentioned above, the method
is the same as in Hacon’s proof of the generic vanishing theorem [Hac04, PP11].
b By Theorem 7.1, it
Proof of Theorem 3.1. Let L be an ample line bundle on A.
suffices to show that

g
−1
) =0
H i A, grF
k M ⊗ R ΨP (Lb → A be the isogeny induced by L. Then, by virtue of ϕL
for i > 0. Let ϕL : A
being étale,


∗ g
−1
g
−1
b ϕ∗L grF)
H i A, grF) ֒→ H i A,
k M ⊗ ϕL R ΨP (Lk M ⊗ R ΨP (LGENERIC VANISHING THEORY VIA MIXED HODGE MODULES

15

is injective, and so we are reduced to proving that the group on the right vanishes
whenever i > 0.
b If (N , F )Let N = ϕ∗L M be the pullback of the mixed Hodge module M to A.
∗
denotes the underlying filtered holonomic D-module, then Fk N = ϕL Fk M because
ϕL is étale. On the other hand, by [Muk81] 3.11b L) ⊗ L.
ϕ∗ Rg ΨP (L−1 ) ≃ H 0 (A,L

We therefore get
b grF N ⊗ L ,b ϕ∗ grF M ⊗ ϕ∗ Rg ΨP (L−1 ) ≃ H 0 (A,b L) ⊗ H i A,
H i A,
Lkk

L

which vanishes for i > 0 by Lemma 8.1.

Going back to the de Rham complex, it is worth recording the complete information one can obtain about grF
k DRA (M) from Saito’s theorem, since this produces
further natural examples of GV-objects on A. One one hand, just as in the proofof Theorem 3.1, we see that each grF
k DRA (M) is a GV-object. On the other hand,
DR
(M)
is
supportedin
non-positive degrees, its Fourier-Mukai transsince grFAkform could a priori have cohomology in negative degrees. The following propositionshows that this is not the case.
Proposition 8.2. If (M, F ) underlies a mixed Hodge module on A, thenbRΦP grF DRA (M) ∈ D≥0 (A).
kcoh

Proof. By a standard application of Serre vanishing (see [PP11, Lemma 2.5]), itb
suffices to show that for any sufficiently positive ample line bundle L on A,0
Hi A, grF
k DR(M, F ) ⊗ R ΨP (L) = 0

for i < 0. Assuming that L is symmetric, R0 ΨP (L) is easily seen to be the dual of
the locally free sheaf Rg ΨP (L−1 ), and so arguing as in the proof of Theorem 3.1,
we are reduced to proving thatb ϕ∗ grF DR(M, F ) ⊗ L−1 = 0
Hi A,
L

k

F
whenever i < 0. But since ϕ∗L grF
k DR(M, F ) ≃ grk DR(N , F ), this is an immediate

consequence of part (2) of Saito’s vanishing Theorem 5.2.

Corollary 8.3. If (M, F ) underlies a mixed Hodge module on A, thenF
grF
k DRA (M) and RHom grk DRA (M), OA [g]are GV-objects on A. Therefore the graded pieces of the de Rham complexes
associated to such D-modules form a class of GV-objects which is closed underGrothendieck duality.

Proof. Note that, by definition, the GV-objects on A are precisely those E ∈Dbcoh (OA ) for which RΦP (E) ∈ c D≤0b), and that the Fourier-Mukai and ducoh (OA
ality functors satisfy the exchange formula(RΦP E)∨ ≃ RΦP −1 (E ∨ )[g]for any object E in Dbcoh (OA ) (see e.g. [PP11] Lemma 2.2). Observing that P and
P −1 differ only by multiplication by −1, the statement follows from the Proposition
above and the equivalence in [Kas04, Proposition 4.3], which says that D≥0b) iscoh (OA
the category obtained by applying the functor RHom(−, OAb) to c D≤0
(O
)
.

b
coh
A

16

M. POPA AND C. SCHNELL

D. Applications to generic vanishing
9. The decomposition theorem for the Albanese map. Let X be a smoothcomplex projective variety of dimension n, let A = Alb(X) be its Albanese variety,and let a : X → A be the Albanese map (for some choice of base point). As before,
we set g = dim A.
To understand the Hodge theory of the Albanese map, we consider the directbH
image f∗ QH
X [n] in D MHM(A). Here QX [n] is the trivial Hodge module on X; its
underlying perverse sheaf is QX [n], and its underlying filtered holonomic D-moduleH
is (OX , F ), where grF
k OX = 0 for k 6= 0. Note that QX [n] ∈ HMn (X) is pure ofweight n, because QX is a variation of Hodge structure of weight 0. According to
the decomposition theorem [Sai88, Théorème 5.3.1 and Corollaire 5.4.8], we have a(non-canonical) decompositionMf∗ QH
Mi [−i],X [n] ≃i∈Z

iwhere each Mi = H f∗ QH
X [n] is a pure Hodge module on A of weight n+i. Each Mi

can be further decomposed (canonically) into a finite sum of simple Hodge modules
M
Mi,j ,Mi =jwhere Mi,j has strict support equal to some irreducible subvariety Zi,j ⊆ A; theperverse sheaf underlying Mi,j is the intersection complex of a local system on a
Zariski-open subset of Zi,j . Note that since f is projective, we have the Lefschetz
isomorphism [Sai90, Théorème 1]
M−i ≃ Mi (i),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 Mi from n + i to n − i, requires
some explanation. If (Mi , F ) denotes the filtered D-module underlying Mi , thenthe filtered D-module underlying Mi (i) is (Mi , F•−i ); thus the above isomorphism
means that Fk M−i ≃ Fk−i Mi .
To relate this decomposition with some concrete information about the Albanesemap, we use Laumon’s formula (Theorem 5.5) to compute the associated graded ofbthe complex of filtered D-modules a∗ (OX , F ) underlying a∗ QH
X [n] ∈ D MHM(A).
To simplify the notation, we let
V = H 0 (X, Ω1X ) = H 0 (A, Ω1A )and S = Sym(V ∗ ).

Proposition 9.1. With notation as above, we have
i
h
•−g
1
•−g+1
n
•−g+n
,
O
⊗
S
→
Ω
⊗
S
→
·
·
·
→
Ω
⊗
S
grF
a
(O
,
F
)
≃
Ra
X
∗
X
∗
X
X
•
with differential induced by the evaluation morphism OX ⊗ H 0 (X, Ω1X ) → Ω1X .
Proof. The cotangent bundle of A is isomorphic to the product A × V , and sousing the notation from §5, we have AA = OA ⊗ S as well as a∗ AA = OX ⊗ S.
Consequently, ωX ⊗ a∗ AA can be resolved by the complexhiA•−n→ Ω1X ⊗ A•−n+1→ · · · → ΩnX ⊗ A•X ⊗ S
XXGENERIC VANISHING THEORY VIA MIXED HODGE MODULES

17

placed as usual in degrees −n, . . . , 0. Applying Theorem 5.5, we find that
i
h
•−g
1
•−g+1
n
•−g+n
,
O
⊗
S
→
Ω
⊗
S
→
·
·
·
→
Ω
⊗
S
grF
a
(O
,
F
)
≃
Ra
X
∗
X
∗
X
X
•
with Koszul-type differential induced by the morphism OX → Ω1X ⊗ V ∗ , which inturn is induced by the evaluation morphism OX ⊗ V → Ω1X .

On the other hand, we know from the decomposition theorem that
M
Mi,j [−i].
a∗ Q HX [n] ≃
i,j

It follows that grF• a∗ (OX , F ) splits, as a complex of graded modules over AA =OA ⊗ S, into a direct sum of modules of the form grF• Mi,j . Putting everything
together, we obtain a key isomorphism which relates generic vanishing, zero sets ofholomorphic one-forms on X, and the topology of the Albanese mapping.
Corollary 9.2. With notation as above, we have
i M
h
grF
Ra∗ OX ⊗ S •−g → Ω1X ⊗ S •−g+1 → · · · → ΩnX ⊗ S •−g+n ≃• Mi,j [−i]i,jin the bounded derived category of graded OA ⊗ S-modules.
Since the Mi,j are Hodge modules on an abelian variety, it follows from Theorem 3.1 that each grF
k Mi,j is a GV-sheaf on A. The isomorphism in Corollary 9.2
shows that whether or not the entire complex grF
k a∗ (OX , F ) satisfies a generic vanishing theorem is determined by the presence of nonzero Mi,j with i > 0. We shallsee in the next section how this leads to a generic vanishing theorem of Nakano-type.
To conclude this section, we recall a well-known condition on the fibers of the
Albanese map that guarantees vanishing of the Mi in a certain range.
Proposition 9.3. For ℓ ∈ N, let Aℓ = y ∈ A dim a−1 (y) ≥ ℓ , and define thedefect of semi-smallness of the map a : X → A to beδ(a) = max 2ℓ − dim X + dim Aℓ .
ℓ∈Ni

Then we have H a∗ QH
X [n] = 0 whenever |i| > δ(a).

To illustrate the meaning of this numerical condition, suppose that X is of maximal Albanese dimension, so that dim X = dim a(X). In this case, 2ℓ − dim X +dim Aℓ = 2ℓ − codim Aℓ , a(X) . We obtain for instance that a is semi-small iff
δ(a) = 0, in which case Mi = 0 for i 6= 0 .
Proof. Since M−i ≃ Mi (i), it suffices to prove that if Mi 6= 0, then i ≤ δ(a).
Suppose that M is one of the simple summands of the Hodge module Mi , whichmeans that M [−i] is a direct factor of a∗ QH
X [n]. The strict support of M is anirreducible subvariety Z ⊆ A, and there is a variation of Hodge structure of weightn + i − dim Z on a dense open subset of Z whose intermediate extension is M . Letz ∈ Z be a general point, and let i : {z} ֒→ A be the inclusion. Then H − dim Z i∗ M
is the corresponding Hodge structure, and so we get H − dim Z i∗ M 6= 0. This implies
that we must haveH − dim Z+i i∗ a∗ QH
X [n] 6= 0

18

M. POPA AND C. SCHNELL

as well. Now let Y = a−1 (z). By the formula for proper base change [Sai90, (4.4.3)],
H
we have i∗ a∗ QH
X [n] ≃ a∗ QY [n], and so we find thatn−dim Z+in−dim Z+i
H − dim Z+i i∗ a∗ QH
a∗ Q H(Y )X [n] ≃ H
Y =H

is the usual mixed Hodge structure on the cohomology of the projective variety
Y . By the above, this mixed Hodge structure is nonzero; it follows for dimension
reasons that n − dim Z + i ≤ 2 dim Y . Consequently, i ≤ δ(a) as claimed.

10. Generic vanishing on the Picard variety. We now address the generic
vanishing theorem of Nakano type, Theorem 3.2 of the introduction, and related
questions. Ideally, such a theorem would say thatonV q (ΩpX ) = L ∈ Pic0 (X) H q X, ΩpX ⊗ L 6= 0

b Unfortunately, such a good statement is
has codimension at least |p + q − n| in A.
not true in general (see [GL87, §3] for an example). To simplify our discussion ofwhat the correct statement is, we make the following definition.
Definition 10.1. Let X be a smooth projective variety. We say that X satisfies
the generic Nakano vanishing theorem with index k if
codim V q (ΩpX ) ≥ |p + q − n| − kfor every p, q ∈ N.
Note that the absolute value is consistent with Serre duality, which implies thatn−p).
V q (ΩpX ) ≃ V n−q (ΩX

We can use the analysis in §9 to obtain a precise formula for the index k. In
particular, Theorem 3.2 is a consequence of Proposition 9.3 and the following result.
Theorem 10.2. Let X be a smooth complex projective variety of dimension n, with
Albanese mapping a : X → A. Definek(X) := max k ∈ Z H k a∗ QH
X [n] 6= 0 .

Then X satisfies the generic Nakano vanishing theorem with index k(X), but notwith index k(X) − 1.

Proof. It follows from the equivalence established in [Pop09, §3] that generic Nakano
vanishing with index k is equivalent to the statement that(10.3)RΦP Ra∗ ΩpX [n − p] ∈ c D≤kb).
coh (OA

We shall prove that this formula holds with k = k(X) by descending induction on
p ≥ 0, starting from the trivial case p = n + 1. To simplify the bookkeeping, we setihC• = Ra∗ OX ⊗ S •−g → Ω1X ⊗ S •−g+1 → · · · → ΩnX ⊗ S •−g+n .
In particular, recalling that (Mi , F ) is the filtered D-module underlying the Hodge
module Mi = H i a∗ QH
X [n], we get from Corollary 9.2 that
h
i M
1
n
n−p
Cg−p = Ra∗ ΩpX → Ωp+1
⊗
S
→
·
·
·
→
Ω
⊗
S
≃
grF
X
g−p Mi [−i].
XiGENERIC VANISHING THEORY VIA MIXED HODGE MODULES

19c
From Theorem 3.1, we know that RΦP (grFb), and so we conclude
g−p Mi ) ∈ Coh(OA≤kcthat RΦP Cg−p ∈ Dcoh (OAb) for 0 ≤ p ≤ n. It is clear from the definition of Cg−pthat there is a distinguished triangle′′
Cg−p→ Cg−p → Ra∗ ΩpX [n − p] → Cg−p[1],′
in which Cg−p
is an iterated extension of Ra∗ ΩrX [n − r] for p + 1 ≤ r ≤ n. From≤k′the inductive hypothesis, we obtain that RΦP Cg−p
∈ c Dcoh(OAb). Now apply thefunctor RΦP to the distinguished triangle to conclude that (10.3) continues to holdfor the given value of p.
This argument can be reversed to show that k(X) is the optimal value for theindex. Indeed, suppose that X satisfies the generic Nakano vanishing theorem withsome index k. Since each complex Cj is an iterated extension of Ra∗ ΩpX [n − p],F
the above shows that RΦP Cj ∈ c D≤kb), and hence that RΦP (grp Mi ) = 0 forcoh (OA
every p ∈ Z and i > k. Because the Fourier-Mukai transform is an equivalence of
categories, we conclude that Mi = 0 for i > k, which means that k(X) ≤ k.


Example 10.4. Our result explains the original counterexample from [GL87, §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, thena : X → A is the Albanese mapping, and a short computation shows that
H 2 (X, Ω3X ⊗ a∗ L) ≃ H 0 (C, ωC ⊗ L) 6= 0
for every L ∈ Pic0 (A). This example makes it clear that the index in the genericNakano 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
QA [4] for k = 0,p k
H a∗ QX [4] ≃ QC [1] for k = ±1,0otherwise.

Thus k(X) = 1 is the correct value for the index in this case.

Theorem 3.2 and its proof improve the previously known generic vanishing resultsfor ΩpX with p < n, and recover those for ωX (or its higher direct images):First, it was proved in [PP11, Theorem 5.11] that
codim V q (ΩpX ) ≥ q + p − n − µ(a),with µ(a) = 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 δ(a) ≤ µ(a).
Secondly, in the case of the lowest nonzero piece of the filtration on Ra∗ (OX , F )the situation is better than what comes out of Theorem 10.2. This allows one
to recover the original generic vanishing theorem for ωX of [GL87], as well as its
extension to higher direct images Rj a∗ ωX given in [Hac04]. Indeed, in the proof
above note thatM
grF
(10.5)
Cg−n = Ra∗ ωX ≃g−n Mi [−i].
i

i

grF
g−n Mi .

This shows that R a∗ ωX ≃Since these sheaves are torsion-free by
virtue of Kollár’s theorem, it follows that grF
g−n Mi = 0 unless 0 ≤ i ≤ dim X −dim a(X). Thus one recovers the original generic vanishing theorem of Green and20

M. POPA AND C. SCHNELL

Lazarsfeld. A similar argument works replacing ωX by higher direct images Ri f∗ ωY ,where f : Y → X is a projective morphism with Y smooth and X projective andgenerically finite over A.
Note. It is worth noting that since for an arbitrary projective morphism f : X → Y
a decomposition analogous to (10.5) continues to hold, one recovers the main resultof [Kol86, Theorem 3.1], namely the splitting of Rf∗ ωX in Dbcoh (Y ). This is ofcourse not a new observation: it is precisely Saito’s approach to Kollár’s theorem
and its generalizations.
E. Cohomological support loci for local systems
11. Supports of transforms. Let A be an abelian variety of dimension g, and set
as before V = H 0 (A, Ω1A ) and S = Sym V ∗ . Let M be a mixed Hodge module on
A, with underlying filtered D-module (M, F ). Then grF• M is a finitely-generated
graded module over Sym ΘA ≃ OA ⊗ S, and we denote the associated coherent
sheaf on T ∗ A = A × V by C (M, F ). We may then define the total Fourier-Mukai
transform of grF• M to be
 MRΦP grF
RΦP (grF
• M =
k M),
k∈Z

which belongs to the bounded derived category of graded modules over OAb ⊗S. The
geometric interpretation is as follows: the object in Dbcoh OA×V
corresponding tobF
the total Fourier-Mukai transform of gr• M isRΦP C (M, F ) = Rp23∗ p∗13 C (M, F ) ⊗ p∗12 P
where the notation is as in the diagram
p13

b×V
A×A

A×V ✛

p12

p23

✲ A
b×V

❄
b
A × A.

Let now X be a smooth projective variety of dimension n, let a : X → A its
Albanese map, and consider again the decomposition
M
Mi,j [−i] ∈ Db MHM(A).
a∗ Q HX [n] ≃
i,j

Denote by Ci,j = C (Mi,j , F ) the coherent sheaf on T ∗ A = A×V determined by theb × V of the total Fourier-Mukai transforms
Hodge module Mi,j . The supports in A
RΦP Ci,j are of a very special kind; this follows by using a result of Arapura [Ara92].
To state the result, we recall the following term coined by Simpson [Sim93, p. 365].
b × H 0 (A, Ω1 ) of the form
Definition 11.1. A triple torus is any subvariety of A
A


0
1
∗
0
1
b × H (A, ΩA ) ,im ϕ : B̂ × H (B, ΩB ) ֒→ A

for a surjective morphism ϕ : A → B to another abelian variety B. A subvariety iscalled a torsion translate of a triple torus if it is a translate of a triple torus by apoint of finite order in A × H 0 (A, Ω1A ).

GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

21

Proposition 11.2. With notation as above, every irreducible component of theb×V.
support of RΦP Ci,j is a torsion translate of a triple torus in A

Proof. It is convenient
to prove a more general statement. For simplicity, denote byC ∈ Dbcoh OA×V the object corresponding to grF• a∗ (OX , F ). Recall from Proposition 9.1 and Corollary 9.2 that we havehi M
Ci,j [−i],(11.3)C ≃ R(a × id)∗ p∗1 OX → p∗1 Ω1X → · · · → p∗1 ΩnX ≃i,jwhere p1 : X × V → X, and the complex in brackets is placed in degrees −n, . . . , 0,and has differential induced by the evaluation map OX ⊗ V → Ω1X .
b×V:
Now define, for m ∈ N, the following closed subsets of Anob × V dim Hk A, L ⊗ C
Z k (m) = (L, ω) ∈ A
≥
m
A×{ω}
n
o

k
b
= (L, ω) ∈ A × V dim H X, L ⊗ K(Ω1X , ω) ≥ m ,where we denote the Koszul complex associated to a single one-form ω ∈ V by
h
i
∧ω
∧ω
∧ω
K(Ω1X , ω) = OX −−→ Ω1X −−→ · · · −−→ ΩnX .

It follows from [Ara92, Corollary on p. 312] and [Sim93] 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 RΦP Ci,j , we also introduceonkb × V dim Hk L ⊗ Ci,j
≥m .
Zi,j
(m) = (L, ω) ∈ A
A×{ω}The decomposition in (11.3) implies that
[\

k−i
Zi,j
µ(i, j) ,Z k (m) =
µ

i,j

where
the union is taken over the set of functions µ : Z2 → N with the property thatPi,j µ(i, j) = m. As in [Ara92, p. 312], this formula implies that each irreduciblekcomponent of Zi,j(m) is also a torsion translate of a triple torus.
Finally, we observe that each irreducible component of Supp RΦP Ci,j mustkkalso be an irreducible component of one the sets Zi,j:= Zi,j(1), which concludes
the proof. More precisely, we have
 [ kSupp RΦP Ci,j =Zi,j .
kk
for all k, with
Indeed, the base change theorem shows that Supp Rk ΦP Ci,j ⊂ Zi,jkequality if k ≫ 0. Now assume that (L, ω) ∈ Zi,j is a general point of a component
k+1,
which is not contained in Supp Rk ΦP Ci,j . We claim that then (L, ω) ∈ Zi,jwhich concludes the proof by descending induction. If this were not the case, thenagain by the base change theorem, the natural map
Rk ΦP Ci,j ⊗ C(L, ω) → Hk A, L ⊗ Ci,j A×{ω}
would be surjective, which would contradict (L, ω) ∈/ Supp Rk ΦP Ci,j .
22

M. POPA AND C. SCHNELL

12. Generic vanishing for rank one local systems. We continue to use thenotation from above. Let Char(X) = Hom(π1 (X), C∗ ) be the algebraic group ofcharacters of X. We are interested in bounding the codimension of the cohomological support lociΣi (X) = ρ ∈ Char(X) H i (X, Cρ ) 6= 0 ,where Cρ denotes the local system of rank one associated to a character ρ. The
structure of these loci has been studied by Arapura [Ara92] and Simpson [Sim93],
who showed that they are finite unions of torsion translates of algebraic subtori ofChar(X). This is established via an interpretation in terms of Higgs line bundles.
(We have already used the more precise result for Higgs bundles from [Ara92] during
the proof of Proposition 11.2.)
We shall assume for simplicity that H 2 (X, Z) is torsion-free (though the argument goes through in general, the only difference being more complicated notation).
In this case, the space of Higgs line bundles on X consisting of pairs (L, θ) of holomorphic line bundles with zero first Chern class and holomorphic one-forms can be
identified as the complex algebraic groupb × V.
Higgs(X) ≃ A

One can define an isomorphism of real (but not complex) algebraic Lie groups
Char(X) → Higgs(X),ρ → (Lρ , θρ ),where θρ is the (1, 0)-part of logkρk interpreted as a cohomology class via theisomorphism H 1 (X, R) ≃ Hom(π1 (X), R).
Note. Lρ is not the holomorphic line bundle Cρ ⊗C OX , except when the characterρ is unitary; this point is stated incorrectly in [Ara92, p.312]. Please refer to[Sim93, p. 364] or page 31 below for the correct description of Lρ .
By means of this identification, the local system cohomology of Cρ can be computed in terms of the Dolbeault cohomology of the Higgs bundle (Lρ , θρ ).
Lemma 12.1 ([Sim92, Lemma 2.2]). There is an isomorphismH k (X, Cρ ) ≃ Hk−n Lρ ⊗ K(Ω1X , θρ )
 h
i
∧θρ
∧θρ
∧θρ
= Hk−n X, Lρ −−→ Lρ ⊗ Ω1X −−→ · · · −−→ Lρ ⊗ ΩnX ,where the Koszul-type complex in brackets is again placed in degrees −n, . . . , 0.
We can now obtain the generic vanishing theorem for local systems of rank one,stated in the introduction.
Proof of Theorem 3.4. Note first that, since Verdier duality gives an isomorphism
H k (X, Cρ ) ≃ H 2n−k (X, C−ρ ),we only need to prove the asserted inequality for k ≥ n. Furthermore, Lemma 12.1
shows that it is enough to prove, for k ≥ 0, the analogous inequality
codimHiggs(X) Z k (1) ≥ 2 k − k(X) ,GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

23

for the subsets Z k (1) ⊆ Higgs(X) that were introduced during the proof of Proposition 11.2. Recall from there that
[
k−i
Zi,j
(1).
Z k (1) =i,jFinally, we saw in the proof of Theorem 13.2 that
codim Supp Rℓ ΦP Ci,j ≥ 2ℓ
for all ℓ. By the same base change argument as before, this is equivalent toℓcodim Zi,j(1) ≥ 2ℓ for all ℓ. Since Ci,j is zero unless i ≤ k(X), we conclude thatk−icodim Z k (1) ≥ min codim Zi,j(1) ≥ 2 k − k(X) ,iwhich is the desired inequality.



13. Duality and perversity for total transforms. In this section, we take acloser look at the behavior of the object C (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.
Lemma 13.1. Let M be a mixed Hodge module on A, let M ′ be its Verdier dual,
and let C (M, F ) and C (M′ , F ) be the associated coherent sheaves on A × V . Then

RHom RΦP C (M, F ), OA×V≃ ι∗ RΦP C (M′ , F ) ,bwhere ι = (−1Ab) × (−1V ).

Proof. Recall that the Grothendieck dual on a smooth algebraic variety X is given
by DX (−) = RHom −, ωX [dim X] . To simplify the notation, set C = C (M, F )and C ′ = C (M′ , F ). Then
Rp23∗ p∗13 C ⊗ p∗12 P
DA×VRΦP C (M, F ) = DA×V
b
b



≃ Rp23∗ p∗13 DA×V C [g] ⊗ p∗12 P −1


∗
∗
≃ Rp23∗ id × id ×(−1V ) (p∗13 C ′ ) ⊗ id ×(−1Ab) × id (p∗12 P ) [2g]∗≃ (−1Ab) × (−1V ) RΦP C ′ [2g] = ι∗ RΦP C (M′ , F ) [2g].
For the first isomorphism we use Grothendieck duality, while for the second we useb × V = 2g, this implies the result.
Theorem 5.4. Since dim A

Now let a : X → A be the Albanese map of a smooth projective variety ofdimension n. Consider the coherent sheaves Ci,j = C (Mi,j , F ) that arise fromb
the decomposition theorem applied to a∗ QH
X [n] ∈ D MHM(A); see Corollary 9.2
for the notation. Many of the results of the previous sections can be stated very
succinctly in the following way, using the second t-structure introduced in §6.
Theorem 13.2. With notation as above, each RΦP Ci,j is a m-perverse coherentb × V . More precisely, the support of the object RΦP Ci,j is a finite union
sheaf on A
of torsion translates of triple tori, subject to the inequality
codim Supp Rk ΦP Ci,j ≥ 2k

for every k ∈ Z. Moreover, the dual objects
RHom RΦP Ci,j , OA×V
b



24

M. POPA AND C. SCHNELL

have the same properties.
b× V → Ab be the first projection. Since we are dealing with sheaves
Proof. Let q : A
of graded modules, the support of the quasi-coherent sheaf M ℓ
R ΦP grFq∗ Rℓ ΦP Ci,j =k Mi,jk∈Z

is the image of Supp Rℓ ΦP Ci,j under the map q. Thanks to Theorem 3.1, eachbRΦP grF
k Mi,j is a c-perverse coherent sheaf on A, so each irreducible componentof the image has codimension at least ℓ. On the other hand, by Proposition 11.2,
the support of Rℓ ΦP Ci,j is a finite union of translates of triple tori. Since a triple
torus is always of the form B̂ × H 0 (B, Ω1B ), we obtain codim B̂ ≥ ℓ, and therefore
codim Supp Rℓ ΦP Ci,j ≥ 2ℓ.
This implies that RΦP Ci,j belongs to m D≤0. Since Verdier duality forbcoh OA×V
mixed Hodge modules commutes with direct images by proper maps, we have
H
H
DA a∗ QH
X [n] ≃ a∗ DX QX [n] ≃ a∗ QX (n)[n],which shows that each module DA (Mi,j ) is again one of the direct factors in thedecomposition of a∗ QH
13.1, the dual complex is thus again of theX [n]. By Lemmam ≤0
as well. It is then clear from the description
same type, hence lies in Dcoh OA×Vbof the t-structure in §6 that both objects actually belong to the heart m Coh(OA×V),
bhence are m-perverse coherent sheaves.
Note. More precisely, each RΦP (Ci,j ) belongs to the subcategory of m Coh(OA×V)
bconsisting of objects whose support is a finite union of translates of triple tori. It is
not hard to see that this subcategory—unlike the category of m-perverse
coherent
sheaves itself—is closed under the duality functor RHom −, OA×V, because eachb
triple torus has even dimension.
14. Perverse coherent sheaves on the space of holomorphic one-forms.
In this section we observe that, in analogy with the method described in [Pop09],
our method also produces natural perverse coherent sheaves on the affine space
V = H 0 (A, Ω1A ), where A is an abelian variety of dimension g. We shall use thefollowing projection maps:
A✛

p

A×V

q

✲ V

b ✛p
A

b×V
A

q

✲ V

b
Lemma 14.1. Let L be
 an ample line bundle on the abelian variety A. For anybobject E ∈ Dcoh OA×V , we have
RHom Rq∗ (p∗ L ⊗ RΦP E), OV ≃ Rq∗ E ′ ⊗ p∗ (−1A )∗ Rg ΨP (L−1 ) ,where we set


E ′ = (id ×(−1V ))∗ RHom E, OA×V [g] .

Proof. This is an exercise in interchanging Grothendieck duality with pushforward
by proper maps and pullback by smooth maps.

We can rephrase Lemma 8.1, which was basically Saito’s version of the Kodairavanishing theorem for mixed Hodge modules on abelian varieties, as follows.

GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

25

Lemma 14.2. Let M be a mixed Hodge module on an abelian variety A, with
underlying filtered D-module (M, F ), and let C (M, F ) be the coherent sheaf on
A × V associated to grF• M. Then for every ample line bundle L on A, one hasRq∗ p∗ L ⊗ C (M, F ) ∈ Coh(OV ).

Proof. V being affine, it suffices to prove that the hypercohomology of the complex
is concentrated in degree 0. But this hypercohomology is equal to M iHi A × V, L ⊗ p∗ C (M, F ) ≃H A, L ⊗ grF
k M ,
k∈Z

which vanishes for i > 0 because of Lemma 8.1.



We can now obtain perverse coherent sheaves on the affine space V by pushing
forward along the projection q : A × V → V .
Proposition 14.3. Let M be a mixed Hodge module on A, with underlying filtered
D-module (M, F ), and let C (M, F ) be the coherent sheaf on A × V associated tobgrF• M. Then for every ample line bundle L on A, one hasRq∗ p∗ L ⊗ RΦP C (M, F ) ∈ c Coh(OV ).
Proof. By Lemma 6.2, it suffices to prove that

RHom Rq∗ p∗ L ⊗ RΦP C (M, F ) , OV ∈ Coh(OV ).

By Lemma 14.1 and Theorem 5.4, this object is isomorphic to
(14.4)
Rq∗ C (M′ , F ) ⊗ p∗ Rg ΨP (L−1 ) ,where C (M′ , F ) is associated to the Verdier dual M ′ = DA M . Now we apply theb → A be the isogeny defined by L. Then the object
usual covering trick. Let ϕL : A
in (14.4) will belong to Coh(OV ) provided the same is true for



bL .
(14.5)
Rq∗ ϕ∗L C (M′ , F ) ⊗ p∗ L ⊗ H 0 A,

Since ϕ∗L C (M′ , F ) comes from the mixed Hodge module ϕ∗L M ′ , this is a consequence of Lemma 14.2.

F. Strong linearity results
15. Fourier-Mukai for D-modules. A stronger version of the results on cohomological support loci in [GL87] was given in [GL91] (see also various extensions in[CH02]). This is the statement (SL) in the introduction; roughly speaking, it statesb is represented bythat the standard Fourier-Mukai transform RΦP OX in Dbcoh (A)b Here we extend this to thea linear complex in a neighborhood of any point in A.
setting of the trivial D-module OX , and consequently to all the Hodge modules
Mi,j appearing in the previous sections. In order to do this, we shall make useof the Fourier-Mukai transform for D-modules on abelian varieties, introduced by
Laumon [Lau96] and Rothstein [Rot96].
We start by setting up some notation. Let A be a complex abelian variety ofdimension g, and let A♯ be the moduli space of algebraic line bundles with flat

26

M. POPA AND C. SCHNELL

connection on A. Note that A♯ naturally has the structure of a quasi-projective
b there is a canonical vector bundle extension
algebraic variety: on A,b × H 0 (A, Ω1 ) → E b → A
b × C → 0,
0→AAA

b × {1} inside E b. The projection
and A♯ is isomorphic to the preimage of AAb
π : A♯ → A,

(L, ∇) 7→ L

b × H 0 (A, Ω1 ); this corresponds to the fact
is thus a torsor for the trivial bundle A
A
that ∇ + ω is again a flat connection for any ω ∈ H 0 (A, Ω1A ). Note that A♯ is agroup under tensor product, and that the trivial line bundle (OA , d) plays the roleof the zero element.
Recall now that Laumon [Lau96] and Rothstein [Rot96] have extended theFourier-Mukai transform to D-modules. Their generalized Fourier-Mukai transform takes bounded complexes of coherent algebraic D-modules on A to bounded
complexes of algebraic coherent sheaves on A♯ ; we briefly describe it following thepresentation in [Lau96, §3], which is more convenient for our purpose. On the product A × A♯ , the pullback P ♯ of the Poincaré bundle P is endowed with a universal
flat connection ∇♯ : P ♯ → Ω1A×A♯ /A♯ ⊗ P ♯ , relative to A♯ . Given any algebraic left
D-module M on A, interpreted as a quasi-coherent sheaf with integrable connection, we consider p∗1 M ⊗ P ♯ on A × A♯ , endowed with the natural tensor productintegrable connection ∇ relative to A♯ . We then define(15.1)

RΦP ♯ (M) := Rp2∗ DR(p∗1 M ⊗ P ♯ , ∇),where DR(p∗1 M ⊗ P ♯ , ∇) is the usual (relative) de Rham complex
i
h
∇
∇
∇
p∗1 M ⊗ P ♯ −→ p∗1 M ⊗ P ♯ ⊗ Ω1A×A♯ /A♯ −→ · · · −→ p∗1 M ⊗ P ♯ ⊗ ΩgA×A♯ /A♯placed in degrees −g, . . . , 0. As all of the entries in this complex are relative to A♯ ,it follows that RΦP ♯ (M) is represented by a complex of algebraic quasi-coherent
sheaves on A♯ . Restricted to coherent D-modules, this is shown to induce an equivalence of categoriesRΦP ♯ : Dbcoh (DA ) → Dbcoh (OA♯ ).
The same result is obtained by a different method in [Rot96, Theorem 6.2].
Note. Since A♯ is not compact, it is essential to consider algebraic coherent sheaves
on A♯ in the above equivalence. On A, this problem does not arise, because thecategory of coherent analytic D-modules on a smooth projective variety is equivalent
to the category of coherent algebraic D-modules by a version of the GAGA theorem.
Now let X be a smooth projective variety with Albanese map a : X → A. By
first pushing forward to A, or equivalently by working with the pullback of (P ♯ , ∇♯ )to X × A♯ , one can similarly defineRΦP ♯ : Dbcoh (DX ) → Dbcoh (OA♯ ).
In this and the following subsections, our goal is to prove the following linearity
property for the Fourier-Mukai transform of the trivial D-module OX (see Definition 20.1 below for the definition of a linear complex over a local ring).

GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

27

Theorem 15.2. Let X be a smooth projective variety of dimension n, and letRΦP ♯ (OX ) ∈ Dbcoh (OA♯ ) be the Fourier-Mukai transform of the trivial D-moduleanon X. Let (L, ∇) ∈ A♯ be an any point, and denote by R = OA
♯ ,(L,∇) the local ringin the analytic topology. Then the stalk RΦP ♯ (OX ) ⊗OX R is quasi-isomorphic toa linear complex over R.
In combination with Proposition 20.2 below, we obtain that every direct summand of RΦP ♯ (OX ) is also isomorphic to a linear complex in an analytic neighborhood of any given point on A♯ .
Corollary 15.3. Suppose that a Hodge module M occurs as a direct factor ofsome H i a∗ QH
X [n]. Let (M, F ) be the filtered holonomic D-module underlying M ,and let RΦP ♯ (M) ∈ Dbcoh (OA♯ ) be its Fourier-Mukai transform. Then the stalkRΦP ♯ (M) ⊗OX R is quasi-isomorphic to a linear complex over R.
The proof of Theorem 15.2 takes up the next four sections. We will in fact provethe more precise version given in Theorem 3.6 in the introduction, by producing thenatural analogue of the derivative complex of [GL91]. As in that paper, the idea isto pull E back to a complex on the universal covering space H 1 (X, C) = H 1 (A, C),
and to use harmonic forms to construct a linear complex that is isomorphic to thepullback in a neighborhood of the preimage of the given point. We will encounterhowever an important additional difficulty, due to the behavior of general harmonic
forms under wedge product.
16. Analytic description. We begin by giving an analytic description of thespace A♯ , and of the Fourier-Mukai transform for D-modules. For the remainderof the discussion, we shall identify H 1 (A, C) with the space of translation-invariant
complex one-forms on the abelian variety A. As complex manifolds, we then haveb ≃ H 1 (A, OA ) H 1 (A, Z),A♯ ≃ H 1 (A, C) H 1 (A, Z)
andA

and this is compatible with the exact sequence

0 → H 0 (A, Ω1A ) → H 1 (A, C) → H 1 (A, OA ) → 0.
In particular, the universal covering space of A♯ is isomorphic to H 1 (A, C). Given atranslation-invariant one-form τ ∈ H 1 (A, C), let τ 1,0 be its holomorphic and τ 0,1 its
anti-holomorphic part. Then the image of τ in A♯ corresponds to the trivial bundle
A × C, endowed with the holomorphic structure given by the operator ∂¯ + τ 0,1 andthe integrable connection given by ∂ + τ 1,0 . We can summarize this by saying that
τ corresponds to the trivial smooth vector bundle A × C, endowed with the smooth
integrable connection dτ = d + τ .
The Albanese mapping a : X → A induces an isomorphism H 1 (A, C) = H 1 (X, C),under which translation-invariant one-forms on A correspond to harmonic one-forms
on X for any choice of Kähler metric. To shorten the notation, we set
W := H 1 (A, C) = H 1 (X, C).
Then a harmonic form τ ∈ W corresponds to the trivial smooth line bundle X × C,
endowed with the smooth integrable connection dτ = d + τ .
We can similarly interpret the pullback of the Poincaré bundle to the complex
manifold X × W . Let C k (X × W/W ) be the sheaf of smooth complex-valued

28

M. POPA AND C. SCHNELL

relative k-forms on X × W that are in the kernel of ∂¯W , meaning holomorphic in
the direction of W , and denote by
C k (X × W/W ) := p2 ∗ C k (X × W/W )its pushforward to W . Let T ∈ C 1 (X × W/W ) denote the tautological relative
one-form on X × W : the restriction of T to the slice X × {τ } is equal to τ . Thenthe pullback of (P ♯ , ∇♯ ) is isomorphic to the trivial smooth bundle X × W × C,
with complex structure given by the operator ∂¯X + ∂¯W + T 0,1 , and with relative
integrable connection given by the operator ∂X + T 1,0 .
This leads to the following analytic description of the Fourier-Mukai transform(similar to [GL91, Proposition 2.3]). Let
D : C k (X × W/W ) → C k+1 (X × W/W )be the differential operator defined by the rule D(α) = dX α + T ∧ α. Using thatdX T = 0, it is easy to see that D ◦ D = 0.
Lemma 16.1. The complex of OW -modules C • (X×W/W ), D is quasi-isomorphic
to the pullback π ∗ RΦP ♯ (OX ), where π : W → A♯ is the universal cover.
Proof. By the definition of the Fourier transform, RΦP ♯ (OX ) is the derived pushforward, via the projection p2 : X × A♯ → A♯ , of the complex


∇♯
∇♯
∇♯
DR(P ♯ , ∇♯ ) = P ♯ −→ Ω1X×A♯ /A♯ ⊗ P ♯ −→ · · · −→ ΩgX×A♯ /A♯ ⊗ P ♯ ,where (P ♯ , ∇♯ ) denotes the pullback of the Poincaré bundle to X × A♯ . Since
π : W → A♯ is a covering map, we thus obtainπ ∗ RΦP ♯ (OX ) ≃ Rp2∗ (id ×π)∗ DR(P ♯ , ∇♯ ) .

As noted above, the pullback of (P ♯ , ∇♯ ) to X × W is isomorphic to the trivial
smooth bundle X × W × C, with complex structure given by ∂¯X + ∂¯W + T 0,1 , andrelative integrable connection given by ∂X + T 1,0 . By a version of the Poincarélemma, we therefore have

(id ×π)∗ DR(P ♯ , ∇♯ ) ≃ C • (X × W/W ), D .
To obtain the desired result, it suffices then to note thatRi p2 ∗ C k (X × W/W ) = 0, for all k and all i > 0.
This follows from a standard partition of unity argument as in [GH78, p. 42].

To prove Theorem
 15.2, it now suffices to show that the stalk of the complex
C • (X × W/W ), D at any given point τ ∈ W is quasi-isomorphic to a linear
complex. Choose a basis e1 , . . . , e2g ∈ H 1 (X, C) for the space of harmonic one-forms
on X, and let z1 , . . . , z2g be the corresponding system of holomorphic coordinates
on W , centered at the point τ . In these coordinates, we have
T =τ+

2g
X

zj ej ∈ C 1 (X × W/W ).

j=1

an
Let R = OW,τbe the analytic local ring at the point τ , with maximal ideal m andresidue field R/m = C. To simplify the notation, we put
C k := C k (X × W/W ) ⊗OW R.

GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

29

Then the R-module C k consists of all convergent power series of the form
X
X
I(1)
I(2g)
α=
αI ⊗ z1 · · · z2g
αI ⊗ z I :=
I

I∈N2g

where αI ∈ Ak (X) are smooth complex-valued k-forms on X, and the summation isover all multi-indices I ∈ N2g . To describe the induced differential in the complex,
we define the auxiliary operators dτ = d + τ andX
e : C k → C k+1 ,e(α) =ej ∧ αI ⊗ zj z I .
I,j

k

k+1

Then each differential D : C → C
is given by the formulaX
X(16.2)
Dα =(dαI + τ ∧ αI ) ⊗ z I +ej ∧ αI ⊗ zj z I = (dτ + e)α.
I

I,j

Note that each R-module in the complex has infinite rank; moreover, in the formulafor the differential D, the first of the two terms is not linear in z1 , . . . , z2g . Our
goal is then to build a linear complex quasi-isomorphic to
C 0 → C 1 → · · · → C 2n−1 → C 2n ,by using harmonic forms with coefficients in the flat line bundle corresponding toτ . The space of dτ -harmonic forms has the advantage of being finite-dimensional;
in addition, any such form α ∈ Ak (X) satisfies dτ α = 0, and hence
Dα =

2g
X

ej ∧ α ⊗ zj .

j=1

This shows that the differential is linear when restricted to the free R-module
generated by the dτ -harmonic forms. The only problem is that we do not obtain a
subcomplex of (C • , D) in this way, because the wedge product ej ∧ α is in general
no longer harmonic. This difficulty can be overcome by constructing a more carefulembedding of the space of dτ -harmonic k-forms into C k , as we now explain.
17. 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 [Sim92]. Let Ak (X) be the space of smooth complex-valued k-forms on
X. Choose a Kähler metric on X, with Kähler form ω ∈ A2 (X), and denote by
L : Ak (X) → Ak+2 (X),

L(α) = ω ∧ α

the associated Lefschetz operator. The metric gives rise to the ∗-operator∗ : Ak (X) → A2n−k (X),
where n = dim X, and the formula(α, β)X =Z

α ∧ ∗βX

defines a Hermitian inner product on the space Ak (X). With respect to these
inner products, the adjoint of L : Ak (X) → Ak+2 (X) is the operator Λ : Ak (X) →Ak−2 (X). Likewise, the adjoint of the exterior derivative d : Ak (X) → Ak+1 (X) isthe operator d∗ : Ak (X) → Ak−1 (X), described more explicitly as d∗ α = − ∗ d ∗ α.
We use the notation d = ∂ + ∂¯ for the decomposition of d by type; thus ∂ maps
(p, q)-forms to (p + 1, q)-forms, and ∂¯ maps (p, q)-forms to (p, q + 1)-forms.

30

M. POPA AND C. SCHNELL

Now fix a holomorphic line bundle L with flat connection ∇. Let Ak (X, L)
denote the space of smooth k-forms with coefficients in L. The theorem of Corlette[Sim92, Theorem 1] shows that there is (up to rescaling) a unique metric on theunderlying smooth bundle that makes (L, ∇) into a harmonic bundle. Together
with the Kähler metric on X, it defines Hermitian inner products on the spaces
Ak (X, L). In the case at hand, the harmonic metric may be described concretely
as follows. According to our previous discussion, there is a harmonic one-formτ ∈ A1 (X), such that L is isomorphic to the trivial smooth bundle X × C, withcomplex structure given by ∂¯ + τ 0,1 and flat connection given by ∂ + τ 1,0 . The
harmonic metric on L is then simply the standard metric on the bundle X × C.
Consequently, we have
Ak (X, L) = Ak (X),and both the ∗-operator and the inner product induced by the harmonic metricagree with the standard ones defined above.
As before, we let dτ = d + τ be the operator encoding the complex structure andflat connection on (L, ∇); concretely,dτ : Ak (X) → Ak+1 (X),

α 7→ dα + τ ∧ α.

Let d∗τ : Ak (X) → Ak−1 (X) be the adjoint of dτ with respect to the inner products,
and let ∆τ = dτ d∗τ + d∗τ dτ be the Laplace operator; it is an elliptic operator of
second order. If we denote byHτk = ker ∆τ : Ak (X) → Ak (X)the space of dτ -harmonic k-forms, then Hτk is finite-dimensional, and Hodge theory
gives us a decomposition
Ak (X) = Hτk (X) ⊕ ∆τ Ak (X),(17.1)

orthogonal with respect to the inner product on Ak (X). Let Hτ : Ak (X) → Hτk be
the orthogonal projection to the space of harmonic forms. It is not hard to see thatany α ∈ Ak (X) can be uniquely written in the form
(17.2)

α = Hτ α + ∆τ Gτ α,kwhere Gτ α ∈ ∆τ A (X) is the so-called Green’s operator. The uniqueness of thedecomposition implies that dτ Gτ = Gτ dτ and d∗τ Gτ = Gτ d∗τ .
Following Simpson, we have a decomposition dτ = ∂τ + ∂¯τ , whereτ 0,1 + τ 1,0
τ 1,0 − τ 0,1+,
2
2
τ 1,0 + τ 0,1
τ 0,1 − τ 1,0+.
∂¯τ = ∂¯ +2
2
The justification for defining these two peculiar operators is that they satisfy theusual Kähler identities (which fail for the naive choice ∂ + τ 1,0 and ∂¯ + τ 0,1 ).
∂τ = ∂ +Theorem 17.3 (Simpson). The following are true:(1) We have ∂τ ∂τ = ∂¯τ ∂¯τ = ∂τ ∂¯τ + ∂¯τ ∂τ = 0.
(2) Let ∂τ∗ and ∂¯τ∗ denote the adjoints of ∂τ and ∂¯τ , respectively. Then thefirst-order Kähler identities∂¯∗ = −i[Λ, ∂τ ],d∗ = i[Λ, ∂¯τ − ∂τ ].
∂ ∗ = i[Λ, ∂¯τ ],τare satisfied.

ττGENERIC VANISHING THEORY VIA MIXED HODGE MODULES

31(3) We have ∂¯τ ∂τ∗ + ∂τ∗ ∂¯τ = ∂τ ∂¯τ∗ + ∂¯τ∗ ∂τ = 0.
(4) The Laplace operator satisfies∆τ = 2(∂τ ∂ ∗ + ∂ ∗ ∂τ ) = 2(∂¯τ ∂¯∗ + ∂¯∗ ∂¯τ ),
τ

τ

τ

τ

and consequently, dτ -harmonic forms are both ∂τ -closed and ∂τ∗ -closed.
(5) We have H∂τ = H ∂¯τ = H∂τ∗ = H ∂¯τ∗ = 0.
(6) The Green’s operator Gτ commutes with ∂τ , ∂¯τ , ∂τ∗ , and ∂¯τ∗ .
Proof. It is easy to see from the definition that (1) holds. The first-order Kähler
identities in (2) are proved in [Sim92, p. 14]; in this simple case, they can also beverified by hand by a calculation on Cn with the Euclidean metric. From this, (3)and (4) follow as in the case of the usual Kähler identities [Sim92, p. 22]. Finally, (5)is a consequence of (4), and (6) follows from the previous results by the uniquenessof the decomposition in (17.2).
Note. The Higgs bundle associated to (L, ∇) is the smooth vector bundle X × C,
with complex structure defined by the operator
τ
∂¯ +

0,1

− τ 1,0,2

and with Higgs field
τ 1,0 + τ 0,1.
2
Note that θ is holomorphic on account of ∂¯τ ∂¯τ = 0. The complex structure on theoriginal flat line bundle is defined by the operator ∂¯ + τ 0,1 , which means that the
two line bundles are different unless τ 0,1 = −τ 1,0 .
θ=Harmonic theory can be used to solve equations involving ∂τ (or any of the otheroperators), as follows. Suppose that we are given an equation of the form ∂τ α = β.
A necessary and sufficient condition for the existence of a solution α is that ∂τ β = 0
and Hτ β = 0. If this is the case, then among all possible solutions, there is a unique
one that is ∂τ∗ -exact, namely 2∂τ∗ Gτ β. In fact, this is the solution of minimal norm.
Note that we can always define α = 2∂τ∗ Gτ β; but since∂τ 2∂τ∗ Gτ β = β − Hτ β − 2∂τ∗ Gτ (∂τ β),

we only obtain a solution to the original equation when Hτ β = 0 and ∂τ β = 0.
This idea will appear again in the construction below.
18. 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 ∂¯τ and Gτ introduced above, whichhold in suitable Sobolev spaces. Since this is standard material in the theory ofpartial differential equations, we only give the briefest possible summary; all the
results that we use can be found, for example, in [Wel08, Chapter IV].
From the Kähler metric on X, we get an L2 -norm on the space Ak (X) of smooth
k-forms, by the formulaZ
α ∧ ∗ᾱ.
kαk20 = (α, α)X =X

It is equivalent to the usual L2 -norm, defined using partitions of unity. Thereis also a whole family of higher Sobolev norms: for α ∈ Ak (X), the m-th order

32

M. POPA AND C. SCHNELL

Sobolev norm kαkm controls the L2 -norms of all derivatives of α of order at mostkm. The Sobolev space Wm(X) is the completion of Ak (X) with respect to theknorm k−km ; it is a Hilbert space. Elements of Wm(X) may be viewed as k-forms α
with measurable coefficients, all of whose weak derivatives of order at most m aresquare-integrable. Here is the first result from analysis that we need.
k
Theorem 18.1 (Sobolev lemma). If α ∈ Wm
(X) for every m ∈ N, then α agrees
almost everywhere with a smooth k-form, and hence α ∈ Ak (X).

The second result consists of a pair of norm inequalities, one for the differential
operator ∂τ∗ , the other for the Green’s operator Gτ .
Theorem 18.2. Let the notation be as above.
(1) There is a constant C > 0, depending on m ≥ 1, such thatk∂τ∗ αkm−1 ≤ C · kαkm
kfor every α ∈ Wm(X) with m ≥ 1.
(2) There is another constant C > 0, depending on m ≥ 0, such thatkGτ αkm+2 ≤ C · kαkm
kfor every α ∈ Wm
(X).

Proof. The inequality in (1) is easy to prove, using the fact that ∂τ∗ is a first-order
operator and X is compact. On the other hand, (2) follows from the open mapping
theorem. To summarize the argument in a few lines, (17.1) is actually derived froman orthogonal decompositionkkWm(X) = Hτk ⊕ ∆τ Wm+2
(X)k(X). It implies that the bounded linear operatorof the Hilbert space Wmkk∆τ : Wm+2(X) ∩ (Hτk )⊥ → Wm
(X) ∩ (Hτk )⊥is bijective; by the open mapping theorem, the inverse must be bounded as well.
Since Gτ is equal to this inverse on (Hτk )⊥ , and zero on Hτk , we obtain the desired
inequality.

19. Construction of the linear complex. We now return to the proof of Theorem 15.2. Recall that, after pullback to the universal covering space W of A♯ , thestalk of the Fourier-Mukai transform of the D-module OX is represented by thecomplex of R-modules
C 0 → C 1 → · · · → C 2n−1 → C 2n ,with differential
D(α) = (dτ + e)α =XIdτ αI ⊗ z I +X

ej ∧ αI ⊗ zj z I .

I,j

Our goal is to show that C • , D is quasi-isomorphic to a linear complex over R.
We begin by constructing a suitable linear complex from the finite-dimensional
spaces of dτ -harmonic forms. Let Hτ• ⊗ R, δ be the complex
Hτ0 ⊗ R → Hτ1 ⊗ R → · · · → Hτ2n−1 ⊗ R → Hτ2n ⊗ R,

GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

33

with differential obtained by R-linear extension fromXHτ (ej ∧ α) ⊗ zj .
δ : Hτk → Hτk+1 ⊗ R,
δ(α) =j

This is indeed a complex, but to show that δ ◦ δ = 0 requires some computation.
Lemma 19.1. We have δ ◦ δ = 0.
Proof. Before we begin the actual proof, let us make a useful observation: namely,that for every α ∈ Ak (X), one hasdτ (ej ∧ α) = d(ej ∧ α) + τ ∧ ej ∧ α = −ej ∧ dα − ej ∧ τ ∧ α = −ej ∧ dτ α,due to the fact that ej is a closed one-form. Now take α ∈ Hτk . ThenHτ (ej ∧ α) = ej ∧ α − ∆τ Gτ (ej ∧ α) = ej ∧ α − dτ d∗τ Gτ (ej ∧ α),because dτ (ej ∧ α) = 0 by the above observation. Consequently,ek ∧ Hτ (ej ∧ α) = ek ∧ ej ∧ α + dτ ek ∧ d∗τ Gτ (ej ∧ α) ,and since Hτ dτ = 0, this allows us to conclude thatXXδ(δα) =
Hτ ek ∧ Hτ (ej ∧ α) ⊗ zj zk =Hτ (ek ∧ ej ∧ α) ⊗ zj zk = 0,
j,k

j,k

which means that δ ◦ δ = 0.

Note that it is clear from the representation of cohomology via harmonic forms
that the complex thus constructed is quasi-isomorphic to the stalk at a point mapping to (L, ∇) of the complex appearing in the statement of Theorem 3.6.
We shall now construct a sequence of maps f k : Hτk → C k , in such a way that,after R-linear extension, we obtain a morphism of complexes f : Hτ• ⊗ R → C • . In
order for the f k to define a morphism of complexes, the identity
Df k (α) = f k+1 (δα)(19.2)

should be satisfied for every α ∈ Hτk . As a first step, we shall find a formal solution
to the problem, ignoring questions of convergence for the time being. Let Ĉ k be
the space of all formal power series
X
X
I(1)
I(2g)
α=
αI ⊗ z I :=αI ⊗ z1 · · · z2g
I

I∈N2g

with αI ∈ Ak (X) smooth complex-valued k-forms. We extend the various operatorsfrom Ak (X) to Ĉ k by defining, for example,
XXdτ α =dτ αI ⊗ z I ,e(α) =ej ∧ αI ⊗ zj z I ,
etc.
I

k

k

I,j

Note that C ⊆ Ĉ is precisely the subspace of those power series that converge insome neighborhood of X × {z = 0}.
To make sure that f k (α) induces the correct map on cohomology, we requirethat Hτ f k (α) = α. Following the general strategy for solving equations with thehelp of harmonic theory, we impose the additional conditions ∂τ f k (α) = 0 and∂¯τ∗ f k (α) = 0. Under these assumptions, (19.2) reduces to∂¯τ f k (α) + ef k (α) = f k+1 (δα).

34

M. POPA AND C. SCHNELL

On account of ∂¯τ∗ f k (α) = 0 and the Kähler identities, we should then havef k (α) = Hτ f k (α) + ∆τ Gτ f k (α) = α + 2 ∂¯τ ∂¯τ∗ + ∂¯τ∗ ∂¯τ Gτ f k (α)
= α + 2∂¯τ∗ Gτ ∂¯τ f k (α) = α − 2∂¯τ∗ Gτ ef k (α) .
This suggests that we try to solve the equation id +2∂¯τ∗ Gτ e f k (α) = α.
Lemma 19.3. For any dτ -harmonic form α ∈ Hτk , the equation(19.4)
id +2∂¯∗ Gτ e β = ατkhas a unique formal solution β ∈ Ĉ . This solution has the property that Hτ β = α,as well as ∂¯τ∗ β = 0 and ∂τ β = 0.
Proof. To solve the equation formally, we write
∞
∞ X
X
X
βI ⊗ z I =
βℓ ,
β=
ℓ=0

ℓ=0 |I|=ℓmaking βℓ homogeneous of degree ℓ in z1 , . . . , z2g . Taking harmonic parts in (19.4),
it is clear that we must have β0 = α and Hτ βℓ = 0 for ℓ ≥ 1; for the rest, theequation dictates thatX
2zj ∂¯τ∗ Gτ ej ∧ βℓ ,(19.5)
βℓ+1 = −jfor every ℓ ≥ 0, which means that there is a unique formal solution β. It is apparent
from the equation that ∂¯τ∗ β = 0, and so to prove the lemma, it remains to show that
we have ∂τ β = 0. Since ∂τ β0 = 0, we can proceed by induction on ℓ ≥ 0. Using
the Kähler identity ∂τ ∂¯τ∗ = −∂¯τ∗ ∂τ , and the fact that ∂τ (ej ∧ βℓ ) = −ej ∧ ∂τ βℓ , wededuce from (19.5) thatXX
2zj ∂¯τ∗ Gτ ej ∧ ∂τ βℓ ,2zj ∂τ ∂¯τ∗ Gτ ej ∧ βℓ = −∂τ βℓ+1 = −jjand so ∂τ βℓ = 0 implies ∂τ βℓ+1 = 0, as required.



The next step is to prove the convergence of the power series defining the solution
to (19.4). For ε > 0, letonX|zj | < ε ,Wε = τ + z ∈ Wjwhich is an open neighborhood of the point τ ∈ W .
Lemma 19.6. There is an ε > 0, such that for all α ∈ Hτk , the formal power series
β = (id +2∂¯∗ Gτ e)−1 α ∈ Ĉ kτconverges absolutely and uniformly on X × Wε to an element of C k (X × Wε /Wε ).
Proof. If we apply the estimates from Theorem 18.2 to the relation in (19.5), we
find that for every m ≥ 1, there is a constant Cm > 0, such thatX
|zj |kβℓ km−1 ≤ Cm ε · kβℓ km−1
(19.7)
kβℓ+1 km ≤ Cmjholds for all ℓ ≥ 1, provided that z ∈ Wε . Given that β0 = α, we conclude thatkβℓ k0 ≤ (C1 ε)ℓ kαk0 .

GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

35

Now choose a positive real number ε < 1/C1 . We then obtain∞Xkβℓ k0 ≤ℓ=0∞X(C1 ε)ℓ kαk0 =

ℓ=0

kαk0
,
1 − C1 εfrom which it follows that β is absolutely and uniformly convergent in the L2 -norm
as long as z ∈ Wε . To prove that β is actually smooth, we return to the originalform of (19.7). It implies that, for any m ≥ 1,∞X

kβℓ km ≤ kαkm + Cm ε ·

ℓ=0

∞
X

kβℓ km−1 .

ℓ=0P
By induction on m ≥ 0, one now easily shows that ℓ kβℓ km converges absolutely
and uniformly on X × Wε for every m ≥ 0; because of the Sobolev lemma, thismeans that β is smooth on X × Wε . Since we clearly have ∂¯W β = 0, it follows thatβ ∈ C k (X × Wε /Wε ).
The preceding lemmas justify definingf k : Hτk → C k ,f k (α) = (id +2∂¯τ∗ Gτ e)−1 α.

It remains to show that we have found a solution to the original problem (19.2).
Lemma 19.8. For every α ∈ Hτk , we have Df k (α) = f k+1 (δα).
Proof. Let β = f k (α), so that (id +2∂¯τ∗ Gτ e)β = α and ∂τ β = 0. Noting that
δ(α) = Hτ (eα), we need to show that(id +2∂¯τ∗ Gτ e)(∂¯τ + e)β = Hτ (eα).
Since e ◦ e = 0 and e(∂¯τ β) = −∂¯τ (eβ), we compute that(id +2∂¯τ∗ Gτ e)(∂¯τ + e)β = ∂¯τ β + eβ + 2∂¯τ∗ Gτ e∂¯τ β = ∂¯τ β + eβ − 2∂¯τ∗ ∂¯τ Gτ eβ.
We always have eβ = Hτ (eβ) + 2∂¯τ ∂¯τ∗ Gτ eβ + 2∂¯τ∗ ∂¯τ Gτ eβ, and so we can simplify
the above to(id +2∂¯τ∗ Gτ e)(∂¯τ + e)β = ∂¯τ β + Hτ (eβ) + 2∂¯τ ∂¯τ∗ Gτ eβ= Hτ (eβ) + ∂¯τ β + 2∂¯∗ Gτ eβ = Hτ (eβ) + ∂¯τ α = Hτ (eβ).
τ

Thus it suffices to prove that Hτ (eβ) = Hτ (eα). But this is straightforward: fromHτ (β) = α and ∂τ β = 0, we get β = α + 2∂τ ∂τ∗ Gτ β, and therefore
eβ = eα − 2∂τ e∂τ∗ Gτ β .
Since Hτ ∂τ = 0, we obtain the desired identity Hτ (eβ) = Hτ (eα).

Note. The decomposition β = α + 2∂τ ∂τ∗ Gτ β is the reason for imposing the additional condition ∂τ f k (α) = 0. Without this, it would be difficult to relate thedτ -harmonic parts of eα and eβ in the final step of the proof.
If we extend R-linearly, we obtain maps of R-modules f k : Hτk ⊗R →C k . Because(19.2) is satisfied, they define a morphism of complexes f : Hτ• ⊗ R, δ → C • , D .
Lemma 19.9. f : Hτ• ⊗ R → C • is a quasi-isomorphism.

36

M. POPA AND C. SCHNELL

Proof. We use the spectral sequence (20.4). The complex Hτ• ⊗ R is clearly linear,and so the associated spectral sequence
1 p,q
E1 = Hτp+q ⊗ Symp (m/m2 ) =⇒ H p+q Hτ• ⊗ R, δdegenerates at E2 by Lemma 20.5. On the other hand, the complex C • also satisfies the conditions neede to define (20.4), giving us a second convergent spectral
sequence with

2 p,q
E1 = H p+q (X, ker ∇) ⊗ Symp (m/m2 ) =⇒ H p+q C • , D ,where ker ∇ is the local system corresponding to (L, ∇). The morphism f induces
a morphism between the two spectral sequences; at E1 , it restricts to isomorphisms
1 p,q
E1 ≃ 2 E1p,q , because Hτk ≃ H k (X, ker ∇). It follows that the second spectral
sequence also degenerates at E2 ; it is then not hard to see that f must be indeed
a quasi-isomorphism.

We have now shown that the complex C • , D is isomorphic, in Dbcoh (R), to alinear complex. This completes the proof of Theorem 15.2.
20. Filtered complexes and linear complexes. This section contains thehomological algebra used in the constructions and proofs of the previous sections.
It reviews and expands some of the content of [LPS10, §1], the main improvementwith respect to that paper being Proposition 20.2.
Definition 20.1. Let (R, m) be a regular local k-algebra of dimension n, with
residue field k = R/m. A linear complex over R is a bounded complex K • , d offinitely generated free R-modules with the following property: there is a system ofparameters t1 , . . . , tn ∈ m, such that every differential of the complex is a matrixof linear forms in t1 , . . . , tn . The property of being linear is obviously not invariant
under isomorphisms. To have a notion that works in the derived category Dbcoh (R),let us say that a complex is quasi-linear over R if it is quasi-isomorphic to a linear
complex over R.
It is an interesting problem to try and find natural necessary and sufficientconditions for quasi-linearity in Dbcoh (R). For instance, if E ∈ Dbcoh (R), then is ittrue that E is quasi-linear over R iff E ⊗R R̂ is quasi-linear over the completion R̂with respect to the m-adic topology?
Here we only treat a special case that was needed in Corollary 15.3., namely thefact that any direct summand in Dbcoh (R) of a quasi-linear complex is again quasilinea, plus a necessary condition for quasi-linearity. The basic tool that we shalluse is the notion of a minimal complex. Recall that a bounded complex (K • , d)of finitely generated free R-modules is called minimal if d(K i ) ⊆ mK i+1 for everyi ∈ Z. This means that every differential of the complex is a matrix with entries
in the maximal ideal m. A basic fact is that every object in Dbcoh (R) is isomorphic
to a minimal complex; moreover, two minimal complexes are isomorphic as objectsof Dbcoh (R) if and only if they are isomorphic as complexes (see [Rob80] for moredetails). Linear complexes are clearly minimal; it follows that a minimal complex
is quasi-linear iff it is isomorphic (as a complex) to a linear complex.
Proposition 20.2. Let E ∈ Dbcoh (R) be quasi-linear. If E ′ is a direct summand of
E in the derived category, then E ′ is also quasi-linear.

GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

37

Proof. Let K • be a linear complex quasi-isomorphic to E, and L• a minimal complex quasi-isomorphic to E ′ . Since E ′ is a direct summand of E in the derived
category, there are morphisms of complexess : L• → K •p : K • → L•andsuch that p ◦ s is homotopic to the identity morphism of L• . Because L• is minimal,
it follows that p ◦ s reduces to the identity modulo m, and is thus an isomorphism.
After replacing p by (p◦ s)−1 ◦ p, we may therefore assume without loss of generalitythat p ◦ s = id.
Now choose a system of parameters t1 , . . . , tn ∈ m with the property that eachdifferential of the complex K • is a matrix of linear forms in t1 , . . . , tn . We are going
to construct a new complex L•0 that has the same property, and is isomorphic toto be the linear partL• . Set Li0 = Li , and define the differential d0 : Li0 → Li+10
of d : Li → Li+1 . That is to say, let d0 be the unique matrix of linear forms in
t1 , . . . , tn with the property that (d − d0 )(Li ) ⊆ m2 Li+1 for every i ∈ Z. It is easy
to see that d0 ◦ d0 = 0, and so (L•0 , d0 ) is a linear complex.
Likewise, we may define s0 : Li0 → K i as the constant part of s : Li → K i ; that is
to say, as the unique matrix with entries in the field k such that (s−s0 )(Li ) ⊆ mK i .
Now consider the commutative diagramLisd✲ Li+1s❄❄d✲ K i+1 .
Ki
By taking linear parts in the identity d ◦ s = s ◦ d, and using the fact that K • isa linear complex, we find that d ◦ s0 = s0 ◦ d0 ; consequently, s0 : L•0 → K • is amorphism of complexes. To conclude the proof, we consider the compositionp ◦ s0 : L•0 → L• .
By construction, p◦s0 reduces to the identity modulo m, and is therefore an isomorphism. This shows that L• is indeed isomorphic to a linear complex, as claimed. 
A necessary condition for quasi-linearity is the degeneration of a certain
 spectral
sequence. To state this, we first recall some general facts. Let K • , F be a filtered complex in an abelian category. We assume that the filtration is decreasing,meaning that F p K n ⊇ F p+1 K n , and satisfies[\
F pK n = K n
and
F p K n = {0}.
p∈Z

p∈Z


Moreover, the differentials should respect the filtration, in the sense that d F p K n ⊆F p K n+1 . Under these assumptions, the filtered complex gives rise to a spectral sequence (of cohomological type)
(20.3)
E1p,q = H p+q F p K • /F p+1 K • =⇒ H p+q K • .

It converges by the standard convergence criterion [McC01, Theorem 3.2].
Going back now to the situation of a regular local k-algebra (R, m) as above, on
any bounded complex K • of R-modules with finitely generated cohomology, we candefine the m-adic filtration by setting
F p K n = mp K n

38

M. POPA AND C. SCHNELL

for all p ≥ 0. Noting that mp /mp+1 ≃ Symp (m/m2 ), we haveF p K n /F p+1 K n ≃ K n ⊗R k ⊗k Symp (m/m2 ).

Provided that each K n has the property that∞\

mp K n = {0},p=1

the filtration satisfies the conditions necessary to define (20.3), and we obtain aconvergent spectral sequence(20.4)
E1p,q = H p+q K • ⊗R k ⊗k Symp (m/m2 ) =⇒ H p+q K • .

It follows from the Artin-Rees theorem that the induced filtration on the limit ism-good; in particular, the completion of H n K • with respect to this filtration isisomorphic to H n K • ⊗R R̂. In this sense, the spectral sequence describes theformal analytic stalk of the original complex.
Lemma 20.5 ([LPS10, Lemma 1.5]). If K • is a linear complex, then the spectral
sequence
E1p,q = H p+q K • ⊗R k ⊗k Symp (m/m2 ) =⇒ H p+q K •degenerates at the E2 -page.

Note that degeneration of the spectral sequence is not a sufficient condition forbeing quasi-linear: over R = kJx, yK, the complex
R




x2
y✲

R⊕2 ,

is an example of a minimal complex that is not isomorphic to a linear complex, butwhere the spectral sequence nevertheless degenerates at E2 .
G. Generalizations and open problems
21. Extensions to D-modules on abelian varieties. In light of our results,
it should be clear that all the theorems in generic vanishing theory are in reality
statements about a certain class of filtered D-modules on abelian varieties, namely
those underlying mixed Hodge modules. Moreover, the dimension (D), linearity(L), and strong linearity (SL) results that we have discussed should be viewed as
properties of the Fourier-Mukai transforms of such D-modules.
A natural question is then whether there is a larger (and more easily described)
class of D-modules on abelian varieties for which the same results are true. It is
known that when (M, F ) underlies a mixed Hodge module M , the D-module M
is always regular and holonomic; when M is pure, M is in addition semi-simple.
This suggests that generic vanishing theory might extend to D-modules with those
properties on abelian varieties.
Using the recent work of Sabbah [Sab05,Sab09] and Mochizuki [Moc07a,Moc07b]
on the correspondence between semi-simple holonomic D-modules and polarizable
twistor D-modules, such an extension is indeed possible. The results are as follows.

GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

39

Theorem 21.1. Let M ∈ Dbh (DA ) be a complex of D-modules with bounded holonomic cohomology on a complex abelian variety A. Then for every k, m ∈ Z, thecohomological support locusonΣkm (M) := (L, ∇) ∈ A♯ dim Hk A, DRA M ⊗OA (L, ∇) ≥ m
is a finite union of translates of triple tori in A♯ ; the translates are by torsion points
when M is of geometric origin. The cohomological support loci satisfy
codim Σkm (M) ≥ 2k

for every k ∈ Z

in the special case when M is a single holonomic D-module.
The theorem implies the analogous result for cohomological support loci of constructible complexes and perverse sheaves, which are now subsets of Char(A); thisis because of the Riemann-Hilbert correspondence. By the usual base change arguments, one derives the following properties of the Fourier-Mukai transform.
Corollary 21.2. For M ∈ Dbh (DA ), the support of the Fourier-Mukai transformRΦP ♯ M ∈ Dbcoh (OA♯ ) is a finite union of translates of triple tori in A♯ ; the translates are by torsion points when M is of geometric origin. When M is a single
holonomic D-module, the system of inequalities
codim Supp Rℓ ΦP ♯ M ≥ 2ℓ

for every ℓ ∈ Z

is satisfied, which implies that RΦP ♯ M is an m-perverse coherent sheaf on A♯ .
For semi-simple holonomic D-modules, there is also a result analogous to (SL).
Theorem 21.3. If M is a semi-simple holonomic D-module on A, then the FourierMukai transform RΦP ♯ M is locally, in the analytic topology, quasi-isomorphic toa linear complex constructed from the cohomology of twists of M.
The proofs will appear elsewhere. They are based on an extension of the FourierMukai transform to twistor D-modules, which produces complexes of coherent analytic sheaves on the twistor space Tw(A♯ ) of the quaternionic manifold A♯ , andon the powerful results about twistor D-modules by Mochizuki and Sabbah. Thisb × H 0 (A, Ω1 ) and A♯ that we usedtreatment unifies the results for the two spaces AAin this paper: they appear as two different fibers of Tw(A♯ ) → P1 . (This can also
be used to deduce that the main result of [GL91] is implied by Theorem 3.6 bypassage to graded pieces.)
22. Open problems. Given the discussion above, a very interesting problem inthis context is the following:
Problem 22.1. Describe classes of D-modules—such as holonomic, regular holonomic, or semi-simple holonomic D-modules—on an abelian variety in terms of
their Fourier-Mukai transforms. In other words, characterize the subcategories of
Dbcoh (OA♯ ) that correspond to the categories of such D-modules under RΦP ♯ .
The two results above give several necessary conditions, so the problem is reallyto find sufficient conditions. Such a description might also shed some light on the
difficult question of which D-modules are of geometric origin.
One may also wonder whether there are extensions of various results in this paperin the non-projective setting.

40

M. POPA AND C. SCHNELL

Problem 22.2. Does the analogue of Theorem 3.1 hold on arbitrary complex tori?
Note that the (SL) type result, Theorem 3.6, generalizes to compact Kähler
manifolds, since the proof only uses harmonic theory. This raises the question
whether our statements of type (D), here relying heavily on vanishing theorems forample line bundles, extend to that context as well.
Problem 22.3. Are there analogues of the generic vanishing theorems 3.2 and 3.4
in the Kähler setting?
References[Ara92] D. Arapura, Higgs line bundles, Green-Lazarsfeld sets, and maps of Kähler manifolds
to curves, Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 2, 310–314.
[AB10] D. Arinkin and R. Bezrukavnikov, Perverse coherent sheaves, Mosc. Math. J. 10 (2010),
no. 1, 3–29.
[BBD82] A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Astérisque 100 (1982),
3–171.
[CH11] J. Chen and C. Hacon, Kodaira dimension of irregular varieties, Invent. Math. 186(2011), no. 3, 481–500.
[CH02] H. Clemens and C. Hacon, Deformations of the trivial line bundle and vanishing theorems, Amer. J. Math. 124 (2002), no. 4, 769–815.
[Del71] P. Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57.
[EL97] L. Ein and R. Lazarsfeld, Singularities of theta divisors and the birational geometry ofirregular varieties, J. Amer. Math. Soc. 10 (1997), no. 1, 243–258.
[GL87] M. Green and R. Lazarsfeld, Deformation theory, generic vanishing theorems, andsome conjectures of Enriques, Catanese and Beauville, Invent. Math. 90 (1987), no. 2,
389–407.
, Higher obstructions to deforming cohomology groups of line bundles, J. Amer.
[GL91]
Math. Soc. 1 (1991), no. 4, 87–103.
[GH78] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, 1978.
[Hac04] C. Hacon, A derived category approach to generic vanishing, J. Reine Angew. Math.
575 (2004), 173–187.
[Kas04] M. Kashiwara, t-structures on the derived categories of holonomic D-modules and coherent O-modules, Moscow Math. J. 4 (2004), no. 4, 847–868.
[Kol86] J. Kollár, Higher direct images of dualizing sheaves II, Ann. of Math. 124 (1986),
171–202.
[KW11] T. Krämer and R. Weissauer, Vanishing theorems for constructible sheaves on abelianvarieties (2011), available at arXiv:1111.4947v1.
[Lau85] G. Laumon, Transformations canoniques et spécialisation pour les D-modules filtrés,
Astérisque 130 (1985), 56–129. Differential systems and singularities (Luminy, 1983).
,
TransformationdeFouriergénéralisée(1996),
availableat[Lau96]arXiv:alg-geom/9603004.
[LPS10] R. Lazarsfeld, M. Popa, and C. Schnell, Canonical cohomology as an exterior module(2010), to appear in the Eckart Viehweg memorial issue of Pure and Applied Math
Quarterly, available at arXiv:1010.5026.
[Lib02] A. Libgober, First order deformations for rank one local systems with a non-vanishingcohomology. Arrangements in Boston: a Conference on Hyperplane Arrangements(1999), Topology Appl. 118 (2002), no. 1-2, 159–168.
[McC01] J. McCleary, A user’s guide to spectral sequences, 2nd ed., Cambridge Stud. Adv. Math.,
vol. 58, Cambridge Univ. Press, Cambridge, 2001.
[Moc07a] T. Mochizuki, Asymptotic behaviour of tame harmonic bundles and an application topure twistor D-modules. I, Mem. Amer. Math. Soc. 185 (2007), no. 869.
[Moc07b], Asymptotic behaviour of tame harmonic bundles and an application to pure
twistor D-modules. II, Mem. Amer. Math. Soc. 185 (2007), no. 870.
b with its application to Picard sheaves,[Muk81] S. Mukai, Duality between D(X) and D(X)
Nagoya Math. J. 81 (1981), 153–175.

GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

41[PP11] G. Pareschi and M. Popa, GV-sheaves, Fourier-Mukai transform, and generic vanishing, Amer. J. Math. 133 (2011), no. 1, 235–271.
[Pop09] M. Popa, Generic vanishing filtrations and perverse objects in derived categories ofcoherent sheaves (2009), to appear in Derived Categories, Tokyo, 2011, available atarXiv:0911.3648.
[Rob80] P. Roberts, Homological invariants of modules over commutative rings, Séminaire deMathématiques Supérieures, vol. 72, Presses de l’Université de Montréal, Montreal,Que., 1980.
[Rot96] M. Rothstein, Sheaves with connection on abelian varieties, Duke Math. J. 84 (1996),
no. 3, 565–598.
[Sab05] C. Sabbah, Polarizable twistor D-modules, Astérisque 300 (2005).
[Sab09], Wild twistor D-modules, Algebraic analysis and around, Adv. Stud. Pure
Math., vol. 54, Math. Soc. Japan, Tokyo, 2009, pp. 293–353.
[Sai88] M. Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), no. 6,
849–995.
, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221–333.
[Sai90][Sai94], On the Theory of Mixed Hodge Modules, Amer. Math. Soc. Transl. Ser. 2 160(1994), 47–61. Translated from Sūgaku.
[Sch11] C. Schnell, Local duality and polarized Hodge modules, Publ. Res. Inst. Math. Sci. 47(2011), no. 3, 705–725.
[Sim92] C. Simpson, Higgs bundles and local systems, Publ. Math. IHES 75 (1992), 5–95.
, Subspaces of moduli spaces of rank one local systems, Ann. Sci. ENS 26 (1993),[Sim93]
361–401.
[Wel08] R. O. Wells Jr., Differential analysis on complex manifolds, 3rd ed., Graduate Texts
in Mathematics, vol. 65, Springer, New York, 2008. With a new appendix by Oscar
Garcia-Prada.