Degenerate Perverse Sheaves on Abelian 3 1 Varieties 0 2 n Rainer Weissauer a J 0 1 ] G A . h 1 Relative generic vanishing t a m LetX beacomplexabelianvariety. Ouraimistoshowthatanirreducibleperverse [ sheaf on X with Euler characteristic zero is translation invariant with respect to 2 someabelian subvarietyofX ofdimension>0. v 7 Notation. LetE(X)denotetheperversesheaveswhoseirreducibleconstituents 4 2 K satisfy c (K)=0. Let N(X) denote negligible perverse sheaves, i.e. those for 2 whichallirreducibleconstituentsare translationinvariantforcertainabeliansub- . 4 varieties of X of dimension >0. Typical examples for translation invariant irre- 20 ducible perverse sheaves are d Xy = Ly [dim(X)], where Ly is the local system on 1 X defined by a character y :p (X,0)→C∗ of the fundamental group of X. Then 1 v: T∗(d y )∼= d y holds for all x ∈ X and d y ∈ N(X). Let F(X) denote the set of ir- x X X X i X reducible perverse sheaves in E(X)\N(X) up to isomorphism. For an arbitrary r perverse sheaf K on X also its character twist Ky =K⊗C Ly is a perverse sheaf, a X and N(X) and E(X) are stable under twisting with y in this sense. Depending on the situation we sometimes write Ky instead of Ky for convenience, e.g. in the cases d Xy =(d X)y . Let M(X) denote perverse sheaves whose irreducible compo- nentsM haveEulercharacteristic c (M)6=0. For isogenies f :X →Y the functors f and f∗ preserve F(X), E(X) and the ∗ categoriesofperversesheaves;an easy consequenceofthepropertiesoftheclass N (defined in [KrW]) and the adjunction formula. A complex K is called Euler negligible,ifitsperversecohomologyisin N(X). An irreducible perverse sheaf K on X is maximal, if for any quotient homo- morphism f :X →BtoasimpleabelianquotientBandgenericcharactertwistsKc of K the direct image Rf∗(Kc ) does not vanish. Let Fmax(X) denote the maximal perverse sheaves in F(X). By corollary 1 below (and the remark thereafter) for K ∈Fmax(X)oneeasilyshowsRf∗(Kc )6=0 foranycharacter c 0. 0 Our main result stated in theorem 4 is the assertion F(X)=0/ respectively the equivalent Theorem 1. For an irreducible perverse sheaf K on a complex abelian variety X with vanishing Euler characteristicc (K)=(cid:229) (−1)idim(Hi(X,K))there exists a i nontrivialabeliansubvarietyA⊆X suchthatT∗(K)∼=K holdsforallx∈A. x Forsimplecomplex abelian varieties this is shown in [KrW]. The main result ofthispaperisthereductionofthetheoremtothecaseofsimpleabelianvarieties. Weremarkthat,ifweassumethecorrespondingresultforsimpleabelianvarieties overthe algebraicclosure k ofa finitefield k , ourproofoftheorem 1 carries over toabelianvarietiesoverk. Infactonestepofourargument(insection8)evenuses methodsofcharacteristic p referring to [W3, appendix]. In contrast,theprooffor the case of simple complex abelian varieties used in [KrW] is of analytic nature, henceunfortunatelycan notbeappliedforfields ofpositivecharacteristic. Reformulationsoftheorem 1. Simpleperversesheaves K onX are oftheform K =i (j E ) for some local system E on an open dense subvariety j:U ֒→Z of ∗ !∗ U U thesupportZ=supp(K). Thesupportisanirreducibleclosedsubvariety i:Z֒→X of X. If the irreducibleperverse sheaf K is translationinvariant withrespect to an abeliansubvarietyAinX,itssupportZ satisfiesZ+A=Z. BytheRiemann-Hilbert correspondence there exists a regular singular holonomic D-module M on X, at- tached to K. The local system E defines an irreducible finite dimensional com- U plex representation f of the fundamental group p (U) ofU. By the A-invariance 1 ofthesingularsupportofK thereexistsanAinvariantclosedsubsetZ′ ofZ,which contains the ramification locus of the perverse sheaf K. In other words, the re- striction of the perverse K to U = Z\Z′ is smooth in the sense that K| = E[d] U holds for a smooth etale sheaf E associated to a representation of the topological fundamentalgroup p (U)ofU whereU can bechosen suchthatU+A=U holds. 1 LetU˜ andZ˜ denotetheimagesofU andZundertheprojectionq:X→X˜ =X/A. Sinceq:X →X˜ =X/Aissmooth,bybasechangetheinducedmorphismq:Z→Z˜ is smooth. The smooth morphism q:Z →Z˜ defines a Serre fibration q:U →U˜. SinceA isconnected, weobtainfromthelongexact homotopysequence p 2(U˜) d // p 1(A) s // p 1(U) // p 1(U˜) //0 . 2 Thefirst map d inthissequence iszero, because p (A)injectsintop (U). Indeed, 1 1 consider the natural group homomorphism r :p (U)→ p (X) induced from the 1 1 inclusion U ֒→X. Obviously the composition r ◦s is the first map of the exact homologysequence 0 // p 1(A) // p 1(X) // p 1(X˜) //0 . hence r ◦s , and therefore s , is injective. To summarize: p (A) is a normal sub- 1 groupofp (U). Weclaimthatp (A)isinthecenterofp (U). Indeed,fora ∈p (A) 1 1 1 1 andg ∈p (U),thereexistsan a ′∈p (A)suchthatgag −1=a ′. Ifweapplytheho- 1 1 momorphism r , this gives r (g )r (a )r (g )−1= r (a ′). Hence r (a )= r (a ′), since p (X)=H (X) is abelian. Therefore a =a ′, because r ◦s is injective. Because 1 1 p (A)isacentralsubgroupof p (U),foranyirreduciblerepresentation f ofp (U) 1 1 1 thereexistsacharacter c ofp (A)suchthatf (ag )=c (a )f (g )holdsfora ∈p (A) 1 1 and g ∈p (U). Since p (X˜) is a free Z-module, any character c of p (A) can be 1 1 1 extendedtoacharacter c ofp (X). Thusc −1⊗f isanirreduciblerepresentation, X 1 X whichis trivialon p (U);in otherwords itisan irreduciblerepresentationof U˜. 1 The last arguments imply that there exists a perverse sheaf K˜ on U˜ such that L(c )−1⊗K = q∗(K˜)[dim(A)] holds on U. Then K˜ necessarily is an irreducible X perverse sheaf. Let K˜ also denote the intermediate extension of K˜ to Z˜, which is an irreducible perverse sheaf on Z˜. Since q : Z → Z˜ is a smooth morphism with connected fibers, the pullback q∗[dim(A)] is a fully faithful functor from the category of perverse sheaves on Z˜ to the category of perverse sheaves on Z. Also L=q∗(K˜)[dim(A)]asperversesheafon Z isstillirreducibleon Z. NowK andLare both irreducible perverse sheaves on Z, whose restrictions on U coincide. Thus K =L. Choosea finiteetalecoveringsuchthat X˜ splits. Then thisimplies(b) =⇒ (c)inthenexttheorem. Theimplications(b)=⇒(c)=⇒(a)areelementary,hence inviewoftheorem 1weget Theorem 2. For an irreducible perverse sheaf K on a complex abelian variety X thefollowingpropertiesareequivalent a) TheEulercharacteristicc (K)vanishes. b) There existsa positivedimensionalabeliansubvarietyA ofX, a translation invariant smooth sheaf L(c ) of rank one on X and and a perverse sheaf K˜ X onX˜ =X/Asuch thatK ∼=L(c )⊗q∗(K˜)[dim(A)]holdsfor thequotientmap X q:X →X˜. 3 c) There exists a finite etale covering of X splitting into a product of two abelian varieties A and X˜, where dim(A)> 0, such that the pullback of K is isomorphic to the external tensor product of a translation invariant per- versesheafon Aand aperversesheafon X˜. Outlineoftheproofoftheorem1. AnyperversesheafK inF(X)hasanassoci- atedD-modulewhosecharacteristicvarietyasasubvarietyofthecotangentbundle isaunionofLagrangians L =L forirreduciblesubvarieties Z ⊆X. Theassump- Z tion c (K)=0 implies that all Z are degenerate. For this see [W] and [KrW]. For simple X therefore Z =X. By the Lagrangian property then L is the zero-section of the cotangent bundle T∗(X). Hence by a well known theorem on D-modules K is attached to a local system defined and smooth on X, and then K is a transla- tion invariant perverse sheaf on X. This proves the statement for simple abelian varietiesand thisessentiallyis theproofof[KrW]. Theauthor’sattemptto givea simple proof along these lines for general abelian varieties was not successful so far, and I wouldliketo thankChristianSchnell forpointing outa gap. For X isogenous to a product A ×A of two simple abelian varieties we use 1 2 methods from characteristic p in the proof. We deal with this case in section 8 after somepreparations in section 6 and 7 buildingon arguments that involvethe tensorcategoriesintroducedin [KrW]. Finally when X has three or more simple factors, we simply use induction on dim(X). The main step, obtained in section 5, requires an analysis of thestalks of primeperversesheavesinthesenseof[W2]. Thesearguments aresheaftheoretic andusespectralsequencesthatnaturallyarise,ifonerestrictsperversesheaveson X to abelian subvarieties (e.g. fibers of homomorphisms used for the induction). Thisisdescribedinsection2andthenisappliedinsection4and5. Acrucialstep insection4 isthereductionto thecase ofperversesheaves inF (X). max An importanttoolforthestudyofhomomorphisms f :X →B in theinduction processwillbethenextlemma. Thislemmacanbeeasilyderivedfromthespecial case of the statement F(A)=0/ where A is a simple abelian variety, whose proof was sketched above. The statement F(A) = 0/ can be converted into a relative generic vanishing theorem for morphisms with simple kernel A in the sense of [KrW]. Factoring an arbitrary homomorphism f : X → B into homomorphisms whosekernelsaresimpleabelianvarieties,aniterativeapplicationoftheassertion F(A)=0/ foreach of thesimpleabelian varieties A defining the sucessivekernels, theneasily givesthenext 4 Lemma 1. Let Abean abeliansubvarietyofX with quotientmap f :X →B=X/A and let K ∈Perv(X,C) be a perverse sheaf on X. Then for a generic character c thedirect imageRf∗(Kc )is aperversesheafon B. Corollary1. H•(X,Fc )=0holdsforF ∈E(X)andgeneric c . Remark. Forfinitelymanyperversesheavesandahomomorphism f :X →B, by lemma 1 one always finds characters c such that G c (K)= pH0(Rf∗(Kc )) is an exact functor on the tensor subcategory T of Db(X,C) generated under the con- c volution product (see [KrW]) by these objects, in the sense that G c maps distin- guishedtrianglestoshortexactsequences. Weusethistoanalysestalks: Suppose the stalk Rf∗(Kc )b vanishes for generic c . Let Fb = f−1(b) be the fiber F = Fb. Then M = (Kc )|F is in pD[−dim(B),0](F) and for generic c all perverse sheaves b Mi = pH−i(M) are acyclic, i.e. H•(F,Mi)=0. Although M and also the perverse sheaves Mi are not necessarily semisimple, this follows from the exactness of the functors G c ; hereas aconsequenceof Hi(F,M) = H0(F,Mi) = H•(F,Mi) for generic c identifying F with A. The same statement carries over to the irre- b ducibleperverseJordan-Ho¨lderconstituents P oftheperversesheaves Mi. Direct images. For the definition of M we fix a suitable generic character c chosen as above. Then Mc = Kcc |F for arbitrary c 0 gives the ‘cohomology’ 0 0 b spectral sequence M H−j(F(b),(Mi)c0)=⇒R−kf∗(Kcc 0)b . j+i=k ThecohomologysheavesH i+k(Mi)= H k(Mi[i])arerelatedtothecohomology i L sheavesH k(M),orequivalentlythestalkcohomologysheavesofthecomplexKc at pointsx∈F viathe‘stalk’spectral sequencewith Ep,q=H p(M−q) b 2 M H −p(Mq) =⇒ H −k(M)=H −k(Kc )|Fb −p−q=−k onF withdifferentials d :H i+k−1(Mi−1)→H i+k+1(Mi). b 2 5 H −d(A)(M0) ... H −d−1(M0) H −d(M0) H −1(M0) H 0(M0) 0 ... H −d(A)(......M... 1) .... H −d−...1(M1) H −d(M1O)OOO.O.O.OOOHOOO−O1O...(OMOOO1O)OOOOH 0(M1) 0000 ............ H −d(A)(Md) ... H −d−1(Md) H −d(Md) ... H −1(Md) H 0(Md) 0 ... ... . . . . . 0 ... ... . . . . . 0 ... H −d(A)(Md(B)) ... H −d−1(Md(B)) H −d(Md(B)) ... H −1(Md(B)) H 0(Md(B)) 0 ... Here d(A) and d(B) are the dimensions of A resp. B. There are edge morphisms H 0(Mi)→H −i(M)and H −i(M)→H −i(M0). Lemma 2. ForsimpleA and f :X →X/AwehaveRf (K)=0 =⇒ K ∈/ F(X). ∗ Proof. Rf∗(K)= 0 implies Rf∗(Kc ) = 0 for generic c . So all Mi are acyclic on A and hence H −j(Mi)=0 for j 6=d(A). Thus by the stalk spectral sequence, H −i−d(A)(Kc )|F(b)∼=H −d(A)(Mi)isatranslationinvariantsheafonAandtherefore is never a skyscraper sheaf. We apply this for the prime component P of K. K According to [W2, lemma 2.1] Rf∗(Kc )=0 for generic c implies Rf∗(PKc )=0 for generic c and K ∈F(X) implies P ∈F(X). Finally for P ∈F(X) the stalk K K H −i(P ) is a skyscraper sheaf at least for one i=n by [W2, lemma 1, part 7]. K K A contradiction. 2 Restriction in steps Considerexact sequences ofabelianvarieties h 0 //B1 // B //B2 //0 p 0 //C // A // B1 // 0 andadiagramofquotienthomomorphismswhere A=g−1(B )and p=g| ,where 1 A C isthekernel oftheprojection g:X →B f X JJJJJgJJJJJJ%% ooooooohooo//ooB77 2=X/A B=X/C 6 Assumption. For perverse K and given quotient morphism f :X →B =X/A 2 supposeforgeneric c R−if∗(Kc ) = 0 , ∀ i<d . ThesevanishingconditionsimplyacyclicityfortheconstituentsP oftheperverse sheavesM0,M1,...,Md−1. Forb ∈B andb ∈B thefibersC∼=F =g−1(b ,b )֒→ f−1(b )=F ∼=A, 2 2 1 1 (b1,b2) 1 2 2 b2 can be identified with A respectively with C up to a translation. This being said, we restrict a generic twist Kc of K (for some generic character c :p 1(X,0)→C∗) tothefiberF b 2 M=M(b2)=(Kc )|F ∈ pD[−dim(B2),0](Fb ); b2 2 thenwefurtherrestrict M to F ֒→F andobtain (b1,b2) b2 N =N(b1,b2):=M(b2)|F(b1,b2) =Kc |F(b1,b2) ∈ pD[−dim(B),0](F(b1,b2)). For Nk = Nk(b ,b ) = pH−k(N) in Perv(F ,C) and Mi = Mi(b ) = pH−i(M) in 1 2 (b,b) 2 1 2 Perv(F ,C) for j=0,..,dim(B ) and i=0,..,dim(B ) there is a ‘doublerestriction’ b 1 2 2 spectral sequence pH−j Mi(b )| =⇒Nk(b ,b ) M (cid:0) 2 F(b1,b2)(cid:1) 1 2 i+j=k Picture. ThefrontrectanglevisualizesthefiberF ,whichisisomorphictoA= b 2 Kern(f),and thefiberF ⊂F isomorphictotheabelian varietyC=Kern(g). (b1,b2) b2 • • OO OO F • (b1,b2) •b 1 |== | B1 B2||||| | | | | | | •oo • C Nowfix b ∈B . Then foralmostallclosed pointsb ∈B theperversesheaf 2 2 1 1 pH0(Md(b )| ) 2 F (b1,b2) 7 is zero, since it defines a perverse quotient sheaf of Md(b ) on F with support in 2 b 2 F . Indeed these supports are disjoint and there are only finitely many con- (b,b ) 1 2 stituents. FurthermoretheperverseconstituentsofthesheavesM0(b ),..,Md−1(b ) 2 2 on F ∼= A are in E(F ), by the vanishing assumption on the direct images: b b 2 2 Rf−i(Kc )=0 fori<d. Forgeneric c wehavethe‘relativecohomological’spectral sequence M H0(F(b1,b2),pH−j(Mi|F(b1,b2)))=⇒R−kg∗(Kc )(b1,b2) . j+i=k Notice i=0,1,...,dim(B ) and j=0,...,dim(B ), where the case j=0 plays a spe- 2 1 cial role as explained above. The spectral sequence is obtained from the double restriction spectral sequence combined with the degenerate cohomology spectral sequenceforgusingH0(F(b,b),Nk)=R−kg∗(Kc )(b,b ) forgeneric c . Nowassume 1 2 1 2 d=dim(B )=m (A)=m (X) . 1 Proposition1. Ford=m (A)=dim(B )=m (X)supposegiven an irreducibleper- 1 verse sheaf K with the vanishing condition R−if∗(Kc )b =0 for i<d and generic 2 c . Thenforfixedb ∈B andgeneric c :p (X,0)→C∗ wehaveanexact sequence 2 2 1 H0(cid:0)F(b1,b2),pH0(Md|F(b1,b2))(cid:1)→R−dg∗(Kc )(b1,b2)→H0(cid:0)F(b1,b2),pH−d(M0|F(b1,b2))(cid:1). In particularR−dg∗(Kc )(b,b) =0 holdsfor almost all b1∈B1 (for fixed b2∈B2), if 1 2 M0 vanishes. NoticeM0=0 iffK doesnot havesupportinthefiber F . b 2 Proof. Lj+i=dH0(F(b1,b2),pH−j(Mi|F(b1,b2)))=⇒R−dg∗(Kc )(b1,b2) forgeneric c and k=d degenerates by Lemma 3 which shows that for 0<i<d we can ignore all terms j =1,...,d−1=dim(B )−1 in this spectral sequence. Since j+i=k, for 1 k=d only the terms (j,i)=(0,d) and the term (j,i)=(d,0) remain. This proves ourassertion. Before we give the proof of the lemma, recall that the abelian variety A can be identified with the ‘front rectangle’ F , the fiber of b for fixed b ∈B , which b 2 2 2 2 containsF (b,b ) 1 2 • • OO OO F • (b1,b2) •b 1 B 1 •oo • C 8 TheirreducibleconstituentsPoftheperversesheavesMi,i<dareacyclicperverse sheaveslivingonthe‘frontrectangle’Aandtheirirreducibleperverseconstituents Pare acyclic. Lemma 3. Suppose m (A)=dim(B )= m (X)= d. Then B is simple and for i= 1 1 0,1,..,d−1and theconstituentsP ofMi forgeneric c we have H•(cid:0)F(b1,b2),pH−j(P|F(b1,b2))c (cid:1) = 0 , ∀ j=0,1,..,d−1 . Proof. TheirreducibleconstituentsP=P˜c (forK|F )oftheMifori=0,1,..,d− 1 are acyclic on A and for p:A→B the direct imagb2e Rp (P) is perverse (c be- 1 ∗ ing generic) so that therefore c (Rp (P))= c (P)=0 holds. Since B is simple of ∗ 1 dimension dim(B )= d, the semisimple perverse sheaf Rp (P) in E(B) is of the 1 ∗ form Rp∗(P)= M my ·d By1 . y ∈Y (c ) Then R−ip (P) = 0 for i 6= d = dim(B ) and all b ∈ B , so that for generic c ∗ b 1 1 1 1 theperversesheaves pH−j(P| ) are acyclicfor j=0,...,−d+1by the remark F(b,b) 1 2 on page 5 applied for (P,p) instead of (K,f). Thus H•(cid:0)F(b1,b2),pH−j(P|F(b1,b2))c (cid:1) vanishesfor j=0,..,d−1. 3 Stalk vanishing conditions Letm (X)betheminimumofthedimensionsofthesimpleabelianvarietyquotients B of X. We say an abelian quotient variety B of X is minimal, if dim(B)= m (X). ForasheafcomplexP on X define m (P) = max{n | H −i(P)=0 forall i<n }. Lemma 4. m (K)≥m (X)holdsforK ∈E(X). Proof by induction on dim(X). Choose f : X → B with simple minimal B. Then Rf∗(Kc ) is perverse for generic c and hence Rf∗(Kc )∈E(B) is of the form Ly my ·d By . By the induction hypothesis we can assume M0 =0, since otherwise the support of K is contained in a proper abelian subvariety of X. M =Kc |f−1(b) has acyclic perverse cohomology sheaves Mi ∈E(Kern(f)) for i=1,...,d−1 and d=m (X). Thenbyinductionm (Mi)≥m (Kern(f))≥m (X)=dimpliesH −n (Mi)=0 for all i=0,...,dim(B)−1=d−1 and all n <d = m (X). Hence m (K)≥d by the stalkspectral sequencediscussedin section1. 9 In the next lemma we givesome information about prime components P of K perversesheaves K ∈F(X). Fordetailsonprimecomponentswereferto[W2]. Lemma 5. ForK ∈F(X)thefollowingholds 1. TheprimecomponentP(K) isin F(X)with n :=m (P )≥m (X). K K 2. K ∈F impliesP ∈F andn =m (P )=m (X). max K max K K 3. For K ∈ F (X) and any minimal quotient B = X/A of X the restriction max M=P | of P to any fiber F = f−1(b) is a complex with Euler perverse K F K cohomology Mi for i=0,..,m (X)−1. For the fiber over the point b=0 fur- thermoreMm (X) has a nontrivialperverse skyscraper quotientconcentrated atthepointzero. Proof. For the first assertion P(K)∈F(X) see [W2, lemma 2.5]. m (P )≥ K m (X) holds by lemma 4 and n := m (P ) and [W2, lemma 1, part 7]. Hence K K n ≥ m (X). By [W2, lemma 4] on the other hand n ≤ m (X) for K ∈ F (X). K K max Hencen =m (P )=m (X)holdsforK∈F (X). Theassertionontheskyscraper K K max subsheafcomesfromtheedgetermoftheabovespectralsequence,sinceforP ∈ K F(X) the cohomology H −n (K)(P ) is a skyscraper sheaf. K ∈ F (X) implies K max PK ∈Fmax(X), since Rf∗(Kc ) is perverse for generic c , and hence Rf∗(Kc )=0 iff Rf∗(PKc )=0by [W2, lemma2.1]. Lemma 6. For K ∈F (X) the support of K is not contained in a translate of a max properabeliansubvarietyA ofX. Proof. For the projection f : X → X/A the support Z of Rf (P ) becomes ∗ K zero: f(Z)={0}. Thisalsoholdsforgenerictwistsof K, sowecouldassumethat Rf (P ) is perverse. Therefore Rf (P ) = 0, since otherwise for a skyscraper ∗ K ∗ K sheaf c (Rf∗PK) > 0 would hold, a contradiction. This implies Rf∗(Kc ) = 0 for generic c contradictingthemaximalityof K. 4 Supports Let A be an abelian subvariety of X and K be an irreducible perverse sheaf on X. Forquotienthomomorphisms f :X →B=X/Aconsidertheassertions 1. K isC-invariantforsomenontrivialabelian subvarietyC ofA. 10