Holonomic D-modules on abelian varieties

45 Pages·2014·0.54 MB·English

HOLONOMIC D-MODULES ON ABELIAN VARIETIES
CHRISTIANSCHNELL
Abstract. WestudytheFourier-MukaitransformforholonomicD-modules
oncomplexabelianvarieties. Amongotherthings,weshowthatthecohomo-
logysupportlociofaholonomicD-modulearefiniteunionsoflinearsubvari-
eties, which go through points of finite order for objects of geometric origin;
thatthestandardt-structureonthederivedcategoryofholonomiccomplexes
corresponds, under theFourier-Mukaitransform, toa certainperverse coher-
ent t-structure in the sense of Kashiwara and Arinkin-Bezrukavnikov; and
thatFourier-MukaitransformsofsimpleholonomicD-modulesareintersection
complexesinthist-structure. ThissupportstheconjecturethatFourier-Mukai
transformsofholonomicD-modulesare“hyperka¨hlerperversesheaves”.
A. Introduction
1. Agenda. In this paper, we begin a systematic study of holonomic D-modules
on complex abelian varieties; recall that a D-module is said to be holonomic if its
characteristicvarietyisaLagrangiansubsetofthecotangentbundle. Regularholo-
nomicD-modules,whichcorrespondtoperversesheavesundertheRiemann-Hilbert
correspondence, are familiar objects in complex algebraic geometry. Due to recent
breakthroughs by Kedlaya, Mochizuki, and Sabbah (summarized in [Sab13]), we
nowhaveanalmostequallygoodunderstandingofirregularholonomicD-modules,
and many important results from the regular case (such as the decomposition the-
orem or the hard Lefschetz theorem) have been extended to the irregular case. A
carefulstudyofanimportantspecialcase,namelythatofcomplexabelianvarieties,
may therefore be of some interest.
The original motivation for this project comes from a sequence of papers by
Green and Lazarsfeld [GL87,GL91], Arapura [Ara92], and Simpson [Sim93]. In
their work on the generic vanishing theorem, these authors analyzed the loci
Sp,q(X)=(cid:8)L∈Pic0(X) (cid:12)(cid:12) dimHq(cid:0)X,Ωp ⊗L(cid:1)≥m(cid:9)⊆Pic0(X),
m X
for X a projective (or compact K¨ahler) complex manifold. Among other things,
theyshowedthateachirreduciblecomponentofSp,q(X)isatranslateofasubtorus
m
byapointoffiniteorder;andtheyobtainedboundsonthecodimensioninthemost
interesting cases (p = 0 and p = dimX). These bounds imply for example that
when the Albanese mapping of X is generically finite over its image, all higher
cohomology groups of ω ⊗L vanish for a generic line bundle L∈Pic0(X).
X
Haconpointedoutthatthecodimensionboundscanbeinterpretedasproperties
of certain coherent sheaves on abelian varieties, and then reproved them using the
2010 Mathematics Subject Classification. Primary14F10;Secondary14K99,32S60.
Key words and phrases. Abelian Variety, Holonomic D-Module, Constructible Complex,
Fourier-Mukai Transform, Cohomology Support Loci, Perverse Coherent Sheaf, Intersection
Complex.
1
2 CHRISTIANSCHNELL
Fourier-Mukai transform [Hac04]. His method applies particularly well to those
coherentsheavesthatoccurintheHodgefiltrationofamixedHodgemodule;based
on this observation, Popa and I generalized all of the results connected with the
generic vanishing theorem (in the projective case) to Hodge modules of geometric
origin on abelian varieties [PS13]. As a by-product, we also obtained results about
certain regular holonomic D-modules on abelian varieties, namely those that can
berealizedasdirectimagesofstructuresheavesofsmoothprojectivevarietieswith
nontrivial first Betti number. A pretty application of the D-module theory to the
geometry of varieties of general type can be found in [PS14].
As we shall see below, all of the results about D-modules in [PS13] remain true
for arbitrary holonomic D-modules on abelian varieties. In most cases, the proof
in the general case turns out to be simpler; we shall also discover that certain
statements – such as the codimension bounds – only reveal their true meaning in
this broader context.
The principal results about holonomic D-modules are summarized in Section 2
to Section 5; for the convenience of the reader, we also translate everything into
the language of perverse sheaves in Section 7. Our main technical tool will be the
Fourier-Mukai transform for algebraic D-modules, introduced by Laumon [Lau96]
and Rothstein [Rot96], and our results suggest a conjecture about the structure
of Fourier-Mukai transforms of (regular or irregular) holonomic D-modules. This
conjecture, together with some evidence for it, is described in Section 6.
2. The structure theorem. Let A be a complex abelian variety, and let D be
A
the sheaf of linear differential operators of finite order. The simplest examples of
left D -modules are line bundles L with integrable connection ∇: L → Ω1 ⊗L.
A A
BecauseAisanabelianvariety,themodulispaceA(cid:92) ofsuchpairs(L,∇)isaquasi-
projective algebraic variety of dimension 2dimA. The basic idea in the study of
D -modules is to exploit the fact that A(cid:92) is so big.
A
One approach is to consider, for a left D -module M, the cohomology groups
A
(inthesenseofD-modules)ofthevarioustwistsM⊗O (L,∇); weusethissymbol
A
to denote the natural DA-module structure on the tensor product M⊗OAL. That
information is contained in the cohomology support loci of M, which are the sets
(2.1) Smk(A,M)=(cid:110)(L,∇)∈A(cid:92) (cid:12)(cid:12)(cid:12) dimHk(cid:16)A,DRA(cid:0)M⊗OA (L,∇)(cid:1)(cid:17)≥m(cid:111).
The definition works more generally for complexes of D-modules; we are especially
interested in the case of a holonomic complex M∈Db(D ), that is to say, a coho-
h A
mologically bounded complex of D -modules with holonomic cohomology sheaves.
A
Our first result is the following structure theorem.
Theorem 2.2. Let M∈Db(D ) be a holonomic complex.
h A
(a) Each Sk(A,M) is a finite union of linear subvarieties of A(cid:92).
m
(b) If M is a semisimple regular holonomic D -module of geometric origin, in
A
the sense of [BBD82, 6.2.4], then these linear subvarieties are arithmetic.
Hereweareusingthenewterm(arithmetic)linearsubvarieties forwhatSimpson
called(torsion)translatesoftripletori in[Sim93,p.365];thedefinitionisasfollows.
Definition 2.3. A linear subvariety of A(cid:92) is any subset of the form
(2.4) (L,∇)⊗im(cid:0)f(cid:92): B(cid:92) →A(cid:92)(cid:1),
HOLONOMIC D-MODULES ON ABELIAN VARIETIES 3
for a surjective morphism of abelian varieties f: A → B with connected fibers,
and a line bundle with integrable connection (L,∇) ∈ A(cid:92). We say that a linear
subvariety is arithmetic if (L,∇) can be taken to be a torsion point.[1]
To prove Theorem 2.2, we use the Riemann-Hilbert correspondence. If M is
a holonomic D -module, then according to a fundamental theorem by Kashiwara
A
[HTT08, Theorem 4.6.6], its de Rham complex
(cid:104) (cid:105)
DR (M)= M→Ω1 ⊗M→···→ΩdimA⊗M [dimA],
A A A
placedindegrees−dimA,...,0,isaperversesheafonA. Moregenerally,DR (M)
A
is a constructible complex for any M∈Db(D ) [HTT08, Theorem 4.6.3], and the
h A
Riemann-Hilbert correspondence [HTT08, Theorem 7.2.1] asserts that the functor
DR : Db (D )→Db(C )
A rh A c A
from regular holonomic complexes to constructible complexes is an equivalence of
categories.
Now let Char(A) be the space of characters of the fundamental group of A; any
character ρ: π (A,0) → C∗ determines a local system C of rank one on A. We
1 ρ
define the cohomology support loci of a constructible complex K ∈Db(C ) as
c A
Smk(A,K)=(cid:110)ρ∈Char(A) (cid:12)(cid:12)(cid:12) dimHk(cid:0)A,K⊗CCρ(cid:1)≥m(cid:111).
Thewell-knowncorrespondencebetweenvectorbundleswithintegrableconnection
and representations of the fundamental group gives a biholomorphic mapping
(2.5) Φ: A(cid:92) →Char(A), (L,∇)(cid:55)→Hol(L,∇),
anditisveryeasytoshow–seeTheorem15.1below–thatthecohomologysupport
loci for M and DR (M) are related by the formula
A
(2.6) Φ(cid:0)A,Sk(A,M)(cid:1)=Sk(cid:0)A,DR (M)(cid:1).
m m A
The proof of Theorem 2.2 is based on the fact that Char(A) and A(cid:92), while
isomorphicascomplexmanifolds,arenotisomorphicascomplexalgebraicvarieties.
According to a nontrivial theorem by Simpson, a closed algebraic subset Z ⊆A(cid:92) is
afiniteunionoflinearsubvarietiesifandonlyifitsimageΦ(Z)⊆Char(A)remains
algebraic [Sim93, Theorem 3.1]. We show (in Theorem 14.6 and Proposition 16.2)
that cohomology support loci are algebraic subsets of A(cid:92) and Char(A); this is
enough to prove the first half of Theorem 2.2. To prove the second half, we show
(in Section 17) that the cohomology support loci of an object of geometric origin
are stable under the action of Aut(C/Q); we can then apply another result by
Simpson, namely that every “absolute closed” subset of A(cid:92) is a finite union of
arithmetic linear subvarieties. Another proof is explained in [Sch13].
3. The Fourier-Mukai transform. A second way to present the information
about the cohomology of twists of M is through the Fourier-Mukai transform for
algebraic D -modules, introduced and studied by Laumon [Lau96] and Rothstein
A
[Rot96]. It is an exact functor
(3.1) FM : Db (D )→Db (O ),
A coh A coh A(cid:92)
defined as the integral transform with kernel (P(cid:92),∇(cid:92)), the tautological line bundle
with relative integrable connection on A×A(cid:92). As shown by Laumon and Roth-
stein, FM is an equivalence between the bounded derived category of coherent
A
4 CHRISTIANSCHNELL
algebraic D -modules, and that of coherent algebraic sheaves on A(cid:92). In essence,
A
thismeansthatanalgebraicD-moduleonanabelianvarietycanberecoveredfrom
the cohomology of its twists by line bundles with integrable connection.
The support of the complex of coherent sheaves FM (M) is related to the co-
A
homology support loci of M: by the base change theorem, one has
(cid:91)
SuppFM (M)= Sk(A,M).
A 1
k∈Z
In particular, the support is a finite union of linear subvarieties. But the Fourier-
Mukai transform of a holonomic complex actually satisfies a much stronger ver-
sion of Theorem 2.2. We shall say that a subset of A(cid:92) is definable in terms of
FM (M) if can be obtained by applying various sheaf-theoretic operations – such
A
as RHom(−,O ), truncation, or restriction to a linear subvariety – to FM (M),
A(cid:92) A
and then taking the support of the resulting complex of coherent sheaves.
Theorem 3.2. Let M∈Db(D ) be a holonomic complex on an abelian variety. If
h A
a subset of A(cid:92) is definable in terms of FM (M), then it is a finite union of linear
A
subvarieties. These linear subvarieties are arithmetic whenever M is a semisimple
regular holonomic D -module of geometric origin.
A
TheproofofTheorem3.2isbasedonananalogueoftheFourier-Mukaitransform
forconstructiblecomplexesK ∈Db(C )(explainedinSection14). Themainpoint
c A
isthatthegroupringR=C[π (A,0)]isarepresentationofthefundamentalgroup,
1
and therefore determines a local system of R-modules L on the abelian variety.
R
BecauseK isconstructibleandp: A→pt isproper,thedirectimageRp∗(K⊗CLR)
therefore belongs to Db (R) and gives rise to a complex of coherent algebraic
coh
sheaves on the affine algebraic variety Char(A) = SpecR. When K = DR (M),
A
we show that the resulting complex of coherent analytic sheaves, pulled back along
Φ: A(cid:92) → Char(A), is canonically isomorphic to FM (M). Both assertions in
A
Theorem 3.2 then follow as before from Simpson’s theorems.
4. Codimension bounds and perverse coherent sheaves. Inequalities for
the codimension of cohomology support loci first appeared in the work of Green
andLazarsfeldonthegenericvanishingtheorem[GL87]. Forexample,whenX isa
projective complex manifold whose Albanese mapping is generically finite over its
image, Green and Lazarsfeld proved that
codimPic0(X)(cid:8)L∈Pic0(X) (cid:12)(cid:12) Hk(X,ωX ⊗L)(cid:54)=0(cid:9)≥k
foreveryk ≥0. Morerecently,Popa[Pop12]noticedthatsuchcodimensionbounds
can be expressed in terms of a certain nonstandard t-structure on the derived
category,introducedbyKashiwara[Kas04]andArinkinandBezrukavnikov[AB10]
in their work on “perverse coherent sheaves”.
In the context of D-modules on abelian varieties, the relationship between codi-
mensionboundsandt-structuresisevencloser. Thefirstresultisthattheposition
of a holonomic complex with respect to the standard t-structure on the category
Db(D ) is detected by the codimension of its cohomology support loci.
h A
Theorem 4.1. Let M∈Db(D ) be a holonomic complex. Then one has
h A
M∈D≤0(D ) ⇐⇒ codimSk(A,M)≥2k for every k ∈Z,
h A 1
M∈D≥0(D ) ⇐⇒ codimSk(A,M)≥−2k for every k ∈Z.
h A 1
HOLONOMIC D-MODULES ON ABELIAN VARIETIES 5
In particular, M is a single holonomic D -module if and only if its cohomology
A
support loci satisfy codimSk(A,M)≥|2k| for every k ∈Z.
1
The natural setting for this result is the theory of perverse coherent sheaves,
developed by Kashiwara and by Arinkin and Bezrukavnikov. As a matter of fact,
there is a perverse t-structure on Db (O ) with the property that
coh A(cid:92)
mD≤co0h(OA(cid:92))=(cid:8)F ∈Dbcoh(OA(cid:92)) (cid:12)(cid:12) codimSuppHkF ≥2k for every k ∈Z(cid:9);
it corresponds to the supporting function m=(cid:4)1codim(cid:5) on the topological space
2
of the scheme A(cid:92), in Kashiwara’s terminology. Its heart mCoh(O ) is the abelian
A(cid:92)
category of m-perverse coherent sheaves (see Section 18).
NowTheorem4.1isaconsequenceofthefollowingbetterresult,whichsaysthat
theFourier-Mukaitransforminterchangesthestandardt-structureonDb(D )and
h A
the m-perverse t-structure on Db (O ).[2]
coh A(cid:92)
Theorem 4.2. Let M∈Db(D ) be a holonomic complex on A. Then one has
h A
M∈D≤k(D ) ⇐⇒ FM (M)∈mD≤k(O ),
h A A coh A(cid:92)
M∈D≥k(D ) ⇐⇒ FM (M)∈mD≥k(O ).
h A A coh A(cid:92)
In particular, M is a single holonomic D -module if and only if its Fourier-Mukai
A
transform FM (M) is an m-perverse coherent sheaf on A(cid:92).
A
The proofs of both theorems can be found in Section 19. The first part of
the argument is to show that when M is a holonomic D -module, the cohomology
A
sheavesHiFM (M)aretorsionsheavesfori>0. Herethecrucialpointisthatthe
A
characteristicvarietyCh(M)insideT∗A=A×H0(A,Ω1)hasthesamedimension
A
as A itself; this makes the second projection
Ch(M)→H0(A,Ω1)
A
finiteoverageneralpointofH0(A,Ω1). TodeduceresultsaboutFM (M),weuse
A A
an extension of the Fourier-Mukai transform to R -modules, where R = R D
A A F A
is the Rees algebra. Choose a good filtration F M, and consider the coherent
•
sheaf grFM on T∗A determined by the graded SymT -module grFM; its support
A •
is precisely Ch(M). The extended Fourier-Mukai transform of the Rees module
R M then interpolates between FM (M) and the complex
F A
(cid:16) (cid:17)
R(p ) p∗ P ⊗p∗ (id×ι)∗grFM ,
23 ∗ 12 13
and because the higher cohomology sheaves of the latter are torsion, we obtain
the result for FM (M). This “generic vanishing theorem” implies also that the
A
cohomologysupportlociSk(A,M)arepropersubvarietiesfork (cid:54)=0; intheregular
1
case, this result is due to Kra¨mer and Weissauer [KW11, Theorem 1.1].
Once the generic vanishing theorem has been established, Theorem 3.2 implies
that HiFM (M) is supported in a finite union of linear subvarieties of lower di-
A
mension; because of the functoriality of the Fourier-Mukai transform, Theorem 4.2
can then be deduced very easily by induction on the dimension.
From there, the basic properties of the m-perverse t-structure quickly lead to
the following result about the Fourier-Mukai transform.
6 CHRISTIANSCHNELL
Corollary 4.3. Let M be a holonomic D -module. The only potentially nonzero
A
cohomology sheaves of the Fourier-Mukai transform FM (M) are
A
H0FM (M), H1FM (M), ..., HdimAFM (M).
A A A
Their supports satisfy codimSuppHiFM (M) ≥ 2i, and if r ≥ 0 is the least
A
integer for which HrFM (M)(cid:54)=0, then codimSuppHrFM (M)=2r.
A A
5. Results about simple holonomic D-modules. According to Theorem 3.2,
the Fourier-Mukai transform of a holonomic D -module is supported in a finite
A
union of linear subvarieties. For simple holonomic D -modules, one can say more:
A
the support of the Fourier-Mukai transform is always irreducible, and if it is not
equaltoA(cid:92),thentheD -moduleis–uptotensoringbyalinebundlewithintegrable
A
connection – pulled back from an abelian variety of lower dimension.
Theorem 5.1. Let M be a simple holonomic D -module. Then
A
SuppFM (M)=(L,∇)⊗im(cid:0)f(cid:92): B(cid:92) →A(cid:92)(cid:1)
A
is a linear subvariety of A(cid:92) (in the sense of Definition 2.3), and we have
M⊗O (L,∇)(cid:39)f∗N
A
for a simple holonomic D -module N with SuppFM (N)=B(cid:92).
B B
The idea of the proof is that for some r ≥ 0, the support of HrFM (M) has
A
to contain a linear subvariety (L,∇)⊗imf(cid:92) of codimension 2r. Because of the
functorialityoftheFourier-Mukaitransform,restrictingFM (M)tothissubvariety
A
(cid:0) (cid:1)
corresponds to taking the direct image f M⊗(L,∇) . We then use adjointness
+
and the fact that M is simple to conclude that M⊗(L,∇) is pulled back from B.
OneapplicationofTheorem5.1istoclassifysimpleholonomicD -moduleswith
A
Eulercharacteristiczero. RecallthattheEulercharacteristicofacoherentalgebraic
D -module M is the integer
A
χ(A,M)=(cid:88)(−1)kdimHk(cid:0)A,DR (M)(cid:1).
A
k∈Z
When M is holonomic, we have χ(A,M) ≥ 0 as a consequence of Theorem 4.2
and the deformation invariance of the Euler characteristic. In the regular case, the
followingresulthasbeenprovedinadifferentwaybyWeissauer[Wei12,Theorem2].
Corollary 5.2. Let M be a simple holonomic D -module. If χ(A,M) = 0, then
A
there exists an abelian variety B, a surjective morphism f: A→B with connected
fibers, and a simple holonomic D -module N with χ(B,N)>0, such that
B
M⊗O (L,∇)(cid:39)f∗N
A
for a suitable point (L,∇)∈A(cid:92).
Now suppose that M is a simple holonomic D-module with H0FM (M) (cid:54)= 0.
A
In that case, the proof of Theorem 5.1 actually gives the stronger inequalities
codimSuppHiFM (M)≥2i+2 for every i≥1.
A
We deduce from this that H0FM (M) is a reflexive sheaf, locally free on the
A
complement of a finite union of linear subvarieties of codimension ≥ 4. This fact
allows us to reconstruct (in Corollary 22.3) the entire complex FM (M) from the
A
locally free sheaf j∗H0FM (M) by applying the functor
A
τ ◦RHom(−,O)◦···◦τ ◦RHom(−,O)◦τ ◦RHom(−,O)◦j .
≤(cid:96)(A)−1 ≤2 ≤1 ∗
HOLONOMIC D-MODULES ON ABELIAN VARIETIES 7
Here (cid:96)(A) is the smallest odd integer ≥ dimA, and j is the inclusion of the open
set where H0FM (M) is locally free.
A
This formula looks a bit like Deligne’s formula for the intersection complex of a
localsystem[BBD82,Proposition2.1.11]. WeinvestigatethisanalogyinSection23,
whereweshowthatthesameformulacanbeusedtodefineanintersectioncomplex
IC (E)∈mCoh(O ),
X X
where j: U (cid:44)→X is an open subset of a smooth complex algebraic variety X with
codim(X \U) ≥ 2, and E is a locally free coherent sheaf on U. This complex has
some of the same properties as its cousin in [BBD82]. In that sense,
FM (M)(cid:39)IC (cid:0)j∗H0FM (M)(cid:1)
A A(cid:92) A
is indeed the intersection complex of a locally free sheaf. When H0FM (M)=0,
A
Theorem 5.1 shows that FM (M) is still the intersection complex of a locally free
A
sheaf, but now on a linear subvariety of A(cid:92) of lower dimension.
6. Aconjecture. Bynow,itwillhavebecomeclearthatFourier-Mukaitransforms
of holonomic D -modules are very special complexes of coherent sheaves on the
A
moduli space A(cid:92). Because the Fourier-Mukai transform
FM : Db (D )→Db (O )
A coh A coh A(cid:92)
is an equivalence of categories, this suggests the following general question.
Question. LetDb(D )denotethefullsubcategoryofDb (D ),consistingofcom-
h A coh A
plexes with holonomic cohomology sheaves. What is the image of Db(D ) under
h A
the Fourier-Mukai transform? In particular, which complexes of coherent sheaves
on A(cid:92) are Fourier-Mukai transforms of holonomic D -modules?
A
In this section, I would like to propose a conjectural answer to this question.
Roughly speaking, the answer seems to be the following:
FM (cid:0)Db(D )(cid:1)=derived category of hyperk¨ahler constructible complexes,
A h A
FM (cid:0)Mod (D )(cid:1)=abelian category of hyperk¨ahler perverse sheaves.
A h A
Recall that the space of line bundles with connection is a hyperk¨ahler manifold:
as complex manifolds, one has A(cid:92) (cid:39) H1(A,C)/H1(cid:0)A,Z(1)(cid:1), and any polarization
of the Hodge structure on H1(A,C) gives rise to a flat hyperk¨ahler metric on A(cid:92).
Here is some evidence for this point of view:
(1) Finite unions of linear subvarieties of A(cid:92) are precisely those algebraic sub-
varieties that are also hyperk¨ahler subvarieties.
(2) Given a holonomic complex M ∈ Db(D ), there is a finite stratification
h A
of A(cid:92) by hyperk¨ahler subvarieties such that the restriction of FM (M) to
A
each stratum has locally free cohomology sheaves.
(3) We prove in Section 20 that a complex of coherent sheaves lies in the sub-
category FM (cid:0)Db(D )(cid:1) if and only if all of its cohomology sheaves do.
A h A
This gives some justification for using the term “constructible complex”.
(4) If we use quaternionic dimension, Theorem 4.2 becomes
dimHSuppHiFMA(M)≤dimHA(cid:92)−i=dimA−i
foraholonomicD -moduleM;thissaysthatthecomplexFM (M)[dimA]
A A
is perverse for the usual middle perversity [BBD82, Chapter 2] over H.
8 CHRISTIANSCHNELL
(5) ForasimpleholonomicD-moduleM,theFourier-MukaitransformFM (M)
A
is the intersection complex of a locally free sheaf.
Unfortunately,nobodyhasyetdefinedacategoryofhyperk¨ahlerperversesheaves,
even in the case of compact hyperk¨ahler manifolds; and our situation presents the
additional difficulty that A(cid:92) is not compact. Nevertheless, I believe that, based on
the work of Mochizuki on twistor D-modules [Moc11], it is possible to make an
educated guess, at least in the case of semisimple holonomic D-modules.
Conjecture 6.1. Let F be a reflexive coherent algebraic sheaf on A(cid:92). Then
there exists a semisimple holonomic D -module M with the property that F (cid:39)
A
H0FM (M) if and only if the following conditions are satisfied:
A
(a) F is locally free on the complement of a finite union of linear subvarieties
of codimension at least 4.
(b) Theresultinglocallyfreesheafadmitsahermitianmetrichwhosecurvature
tensor Θ is SU(2)-invariant and locally square-integrable on A(cid:92).
h
(c) The pointwise norm of Θ , taken with respect to h, is in O(cid:0)d−(1+ε)(cid:1), where
h
d is the distance to the origin in A(cid:92).
Moreover, M is regular if and only if the pointwise norm of Θ is in O(d−2).
h
Thereisacertainamountofredundancyintheconditions. Infact,wecouldstart
from a holomorphic vector bundle E on the complement of a finite union of linear
subvarieties of codimension ≥ 2, and assume that it admits a hermitian metric h
for which (b) and (c) are true. Then h is admissible in the sense of Bando and Siu
[BS94], and E therefore extends uniquely to a reflexive coherent analytic sheaf on
A(cid:92);byvirtueof(c),theextensionisacceptableinthesenseof[Moc11,Chapter21],
and therefore algebraic.[3] In particular, E itself is algebraic, and the discussion at
the end of Section 5 shows that the simple holonomic D -module must be
A
(cid:16) (cid:17)
FM−1 IC (E) ,
A A(cid:92)
theinverseFourier-MukaitransformoftheintersectioncomplexofE. Theproblem
is, of course, to show that this is indeed a simple holonomic D -module.
A
The paper [Moc13] establishes a result equivalent to Conjecture 6.1 in the case
of elliptic curves. The reason for believing that regularity should correspond to
quadratic decay in the curvature is the work of Jardim [Jar02]. In general, the
existence of the metric, and the SU(2)-invariance of its curvature, should be con-
sequences of the fact that every simple holonomic D-module lifts to a polarized
wildpuretwistorD-module. Theremainingpointswillprobablyrequireadditional
methods from analysis. Note that the conjecture is consistent with the result (in
Corollary 25.3) that all Chern classes of FM (M) are zero in cohomology.
A
Another interesting question is whether the existence of the metric in (b) is
equivalent to an algebraic condition such as stability. If that was the case, then
I would guess that the semistable objects are what corresponds to Fourier-Mukai
transforms of not necessarily simple holonomic D -modules.
A
7. Results about perverse sheaves. For the convenience of those readers who
are more familiar with constructible complexes and perverse sheaves, we shall now
translateourmainresultsintothatlanguage. Inthesequel,aconstructiblecomplex
on the abelian variety A means a complex K of sheaves of C-vector spaces, whose
HOLONOMIC D-MODULES ON ABELIAN VARIETIES 9
cohomology sheaves HiK are constructible with respect to an algebraic stratifica-
tionofA,andvanishforioutsidesomeboundedinterval. WedenotebyDb(C )the
c A
boundedderivedcategoryofconstructiblecomplexes. Itisabasicfact[HTT08,Sec-
tion4.5]thatthehypercohomologygroupsHi(A,K)arefinite-dimensionalcomplex
vector spaces for any K ∈Db(C ).
c A
Now let Char(A) be the space of characters of the fundamental group; it is also
themodulispaceforlocalsystemsofrankone. Foranycharacterρ: π (A,0)→C∗,
1
we denote the corresponding local system on A by the symbol C . It is easy to see
ρ
that K⊗CCρ is again constructible for any K ∈Dbc(CA); we may therefore define
the cohomology support loci of K ∈Db(C ) to be the subsets
c A
(7.1) Smk(A,K)=(cid:110)ρ∈Char(A) (cid:12)(cid:12)(cid:12) dimHk(cid:0)A,K⊗CCρ(cid:1)≥m(cid:111),
for any pair of integers k,m ∈ Z. Since the space of characters is very large – its
dimensionisequalto2dimA–theselocishouldcontainalotofinformationabout
the original constructible complex K, and indeed they do.
Our first result is a structure theorem for cohomology support loci.
Definition 7.2. A linear subvariety of Char(A) is any subset of the form
(cid:0) (cid:1)
ρ·im Char(f): Char(B)→Char(A) ,
forasurjectivemorphismofabelianvarietiesf: A→B withconnectedfibers, and
a character ρ∈Char(A). We say that a linear subvariety is arithmetic if ρ can be
taken to be torsion point of Char(A).
Theorem 7.3. Let K ∈Db(C ) be a constructible complex.
c A
(a) Each Sk(A,K) is a finite union of linear subvarieties of Char(A).
m
(b) IfK isasemisimpleperversesheafofgeometricorigin[BBD82,6.2.4],then
these linear subvarieties are arithmetic.
Proof. For (a), we use the Riemann-Hilbert correspondence to find a regular holo-
nomiccomplexM∈Db (D )withDR (M)(cid:39)K. SinceSk(A,K)=Φ(cid:0)Sk(A,M)(cid:1)
rh A A m m
by Theorem 15.1, the assertion follows from Theorem 2.2. The statement in (b)
can be deduced from Theorem 17.2 by a similar argument. (cid:3)
The next result has to do with the codimension of the cohomology support loci.
Recall that the category Db(C ) has a nonstandard t-structure
c A
(cid:16) (cid:17)
πD≤0(C ),πD≥0(C ) ,
c A c A
called the perverse t-structure, whose heart is the abelian category of perverse
sheaves[BBD82]. Weshowthatthepositionofaconstructiblecomplexwithrespect
to this t-structure can be read off from its cohomology support loci.
Theorem 7.4. Let K ∈Db(C ) be a constructible complex. Then one has
c A
K ∈πD≤0(C ) ⇐⇒ codimSk(A,K)≥2k for every k ∈Z,
c A 1
K ∈πD≥0(C ) ⇐⇒ codimSk(A,K)≥−2k for every k ∈Z.
c A 1
Thus K is a perverse sheaf if and only if codimSk(A,K)≥|2k| for every k ∈Z.
1
10 CHRISTIANSCHNELL
Proof. LetM∈Db (D )bearegularholonomiccomplexsuchthatK (cid:39)DR (M).
rh A A
Since Sk(A,K) = Φ(cid:0)Sk(A,M)(cid:1), the first assertion is a consequence of Theo-
m m
rem 19.1. Now let D : Db(C )→Db(C ) be the Verdier duality functor; then
A c A c A
Sk(A,K)=(cid:104)−1 (cid:105)S−k(A,D K)
m Char(A) m A
by Verdier duality. Since K ∈ πD≥0(C ) if and only if D K ∈ πD≤0(C ), the
c A A c A
second assertion follows. The final assertion is clear from the definition of perverse
sheaves as the heart of the perverse t-structure on Db(C ). (cid:3)
c A
Aconsequenceisthefollowing“genericvanishingtheorem”forperversesheaves;
a similar – but less precise – statement has been proved some time ago by Kr¨amer
and Weissauer [KW11, Theorem 1.1].
Corollary 7.5. Let K ∈Db(C ) be a perverse sheaf on a complex abelian variety.
c A
Then the cohomology support loci Sk(A,K) are finite unions of linear subvarieties
m
of Char(A) of codimension at least |2k|. In particular, one has
Hk(cid:0)A,K⊗CCρ(cid:1)=0
for general ρ∈Char(A) and k (cid:54)=0.
The generic vanishing theorem implies that the Euler characteristic
χ(A,K)=(cid:88)(−1)kdimHk(cid:0)A,K(cid:1)
k∈Z
of a perverse sheaf on an abelian variety is always nonnegative, a result originally
duetoFraneckiandKapranov[FK00,Corollary1.4]. Indeed,fromthedeformation
invariance of the Euler characteristic, we get
χ(A,K)=χ(cid:0)A,K⊗CCρ(cid:1)=dimH0(cid:0)A,K⊗CCρ(cid:1)≥0
forageneralcharacterρ∈Char(A). Forsimple perversesheaveswithχ(A,K)=0,
we have the following structure theorem [Wei12, Theorem 2].
Theorem 7.6. Let K ∈Db(C ) be a simple perverse sheaf. If χ(A,K)=0, then
c A
there exists an abelian variety B, a surjective morphism f: A→B with connected
fibers, and a simple perverse sheaf K(cid:48) ∈Db(C ) with χ(B,K(cid:48))>0, such that
c B
K (cid:39)f∗K(cid:48)⊗CCρ
for some character ρ∈Char(A).
Proof. This again follows from the Riemann-Hilbert correspondence and the anal-
ogous result for simple holonomic D -modules in Corollary 5.2. (cid:3)
A
8. Acknowledgements. This work was supported by the World Premier Inter-
national Research Center Initiative (WPI Initiative), MEXT, Japan, and by NSF
grantDMS-1331641. IthankMihneaPopaandPierreSchapirafortheircomments
about the paper, and Takuro Mochizuki, Kiyoshi Takeuchi, Giovanni Morando,
andKentaroHoriforusefuldiscussions. Ananonymousrefereepointedoutseveral
small mistakes, and his/her insightful remarks also suggested a better proof for
Theorem 15.2. Lastly, I am very grateful to my parents-in-law for their hospitality
while I was writing the first version of this paper.