Generic vanishing in characteristic p>0 and the characterization of ordinary abelian varieties
aa r X i v : . [ m a t h . AG ] F e b GENERIC VANISHING IN CHARACTERISTIC p > AND THECHARACTERIZATION OF ORDINARY ABELIAN VARIETIES
CHRISTOPHER D. HACON AND ZSOLT PATAKFALVI
Abstract.
We prove a generic vanishing type statement in positive charac-teristic and apply it to prove positive characteristic versions of Kawamata’stheorems: a characterization of smooth varieties birational to ordinary abelianvarieties and the surjectivity of the Albanese map when the Frobenius stableKodaira dimension is zero. Introduction
Varieties of maximal Albanese dimension.
Let X be a smooth projec-tive variety over an algebraically closed field k . If X admits a generically finitemorphism to an abelian variety f : X → A then we say that X has maximalAlbanese dimension (m.A.d.). The geometry of complex projective m.A.d. vari-eties is extremely well understood. In particular it is known that these varietiesadmit a good minimal model (cf. [Fujino09]), and that the 4-th pluricanonicalmap gives the Iitaka fibration (cf. [JLT11]). The main tool used in provingresults about the geometry of m.A.d. varieties are the generic vanishing theo-rems first developed by Green and Lazarsfeld (and further sharpened by Chen,Hacon, Popa, Pareschi, Schnell, Simpson and others). On the other hand, inpositive characteristic very little is known about generic vanishing and m.A.d.varieties. By a result of the first author and Kov´acs it is known that the obvi-ous positive characteristic version of the generic vanishing theorem does not hold[HK12]. Here we present a generic vanishing statement in positive characteristicwhich, by the results of [HK12] is necessarily weaker than the characteristic zerostatements, but it is strong enough to imply the following positive characteristicversion of the celebrated results of Kawamata [Kawamata81]. Theorem 1.1.1.
Let X be a smooth projective variety over an algebraically closedfield k of characteristic p > , and let a : X → A be the Albanese morphism. (1) If κ S ( X ) = 0 , then a is surjective and in particular we obtain the upperbound b ( X ) ≤ X for the first Betti number. (2) X is birational to an ordinary abelian variety if and only if p ∤ deg a , κ S ( X ) = 0 and b ( X ) = 2 dim( X ) . Here b ( X ) is the first Betti-number, which by definition is dim Q l H ( X, Q l )for any l = p , and κ S ( X ) is the Frobenius stable Kodaira dimension (see the later parts of the introduction or the beginning of § κ S ( X )).Note also that it is well known that b ( X ) = 2 dim A [Liedtke09, page 14].1.2. Generic vanishing over the complex numbers.
To explain our positivecharacteristic generic vanishing statement, let us recall briefly the known resultson generic vanishing over the complex numbers. Let X be a smooth projectivevariety over C of dimension d , let a : X → A be the Albanese morphism and V i ( ω X ) := { P ∈ Pic ( X ) | h i ( X, ω X ⊗ P ) = 0 } the cohomology support loci. Theorem 1.2.1. [GL91, Simpson93]
Every irreducible component of V i ( ω X ) isa (torsion) translate of a (reduced) subtorus of Pic ( X ) of codimension at least i − (dim( X ) − dim( a ( X ))) . If dim( X ) = dim a ( X ) then there are inclusions: V ( ω X ) ⊃ V ( ω X ) ⊃ . . . ⊃ V dim( X ) ( ω X ) = {O X } . In particular if X is a variety of m.A.d., then the V i ( ω X ) have codimension ≥ i ,which implies the vanishing of H i ( X, ω X ⊗ P ) for i > P ∈ Pic ( X ).Hence this result is known as a ”generic vanishing theorem”. It implies thatfor m.A.d. varieties one can replace the ample line bundle in the statement ofKodaira vanishing by a general topologically trivial line bundle. In fact, (1.2.1)can be thought of as the limit of Kodaira vanishing if one considers the alternativepoint of view of [Hacon04] and [PP09]. In these articles the following seeminglyunrelated conjecture of Green and Lazarsfeld was proven. Theorem 1.2.2. [Hacon04, PP09]
Let L be the Poincar´e line bundle on A × ˆ A ,where ˆ A := Pic ( A )( ∼ = Pic ( X )) is the dual abelian variety of A . If the Albaneseimage a ( X ) ⊂ A has dimension d − k , then R i p ˆ A, ∗ ( L X ) = 0 for i [ d − k, d ] and L X := ( a × id ˆ A ) ∗ L . It turns out that in fact (1.2.2) and (1.2.1) are equivalent by [PP11]. To explainthis, note that by applying standard derived category machinery (Grothendieckspectral sequence, projection formula and Grothendieck duality), one can showthat Rp ˆ A, ∗ ( L X ) ∼ = R ˆ S ( D A ( Ra ∗ ω X [ d ])), where D A (?) ∼ = R H om (? , O A [ g ]), g =dim A and R ˆ S : D ( A ) → D ( ˆ A ) is the Fourier-Mukai functor defined in [Mukai81]so that R ˆ S (?) = Rp ˆ A, ∗ ( Lp ∗ A (?) ⊗ L ). Further, by a famous result of Koll´ar, incharacteristic zero Ra ∗ ω X = P kj =0 R j a ∗ ω X [ − j ] (cf. [Kol86]). Thus, (1.2.2) isequivalent to the vanishing H i ( R ˆ S ( D A ( R j a ∗ ω X ))) = 0for each 0 ≤ j ≤ k and i < j − k .The latter condition was shown in [PP11] to be equivalent to (1.2.1). Let φ L :ˆ A → A be the corresponding isogeny determined by the formula φ L ( x ) = t ∗ x L ⊗ L ∨ , ENERIC VANISHING AND ORDINARY ABELIAN VARIETIES 3 where t x is the translation by x ∈ ˆ A . Then the dual RS ( L ) = H ( RS ( L )) = ˆ L of L is a vector bundle on A of rank h ( L ) such that φ ∗ L ( ˆ L ) = L h ( L ) L ∨ (here RS : D ( ˆ A ) → D ( A ) is the inverse Fourier-Mukai functor RS (?) = Rp A, ∗ ( Lp ∗ ˆ A (?) ⊗ L )). Theorem 1.2.3. [PP11, Theorem A]
Let A be an abelian variety over an alge-braically closed field and F a coherent sheaf on A and l ≥ an integer, then thefollowing are equivalent: (1) codim (cid:16) V i ( F ) := { P ∈ ˆ A | h i ( F ⊗ P ) = 0 } (cid:17) ≥ i − l (2) H i ( R ˆ S ( D A ( F ))) = 0 for all i < − l . (3) H i ( A, F ⊗ ˆ L ∨ ) = 0 for i > l and any sufficiently ample line bundle L . Heuristically, we think of ˆ L ∨ as an ample vector bundle which plays the roleof L ) L , so that as deg( L ) increases, ˆ L ∨ corresponds to a smaller and smallermultiple of an ample line bundle (alternatively the rank of ˆ L ∨ increases but h ( ˆ L ∨ ) = 1 is fixed). In this way we interpret, as hinted earlier, the GenericVanishing Theorem as a limit of the Kodaira vanishing theorem. In fact, incharacteristic 0, it is easy to see that Koll´ar vanishing implies that each R j a ∗ ω X satisfies (3) of (1.2.3) with l = 0 and hence that (1.2.2) and (1.2.1) hold (cf.[Hacon04]).1.3. Generic vanishing in positive characteristic.
It is a natural questionto generalize these important results to positive characteristic. The main issuein doing so is that in characteristic zero, Koll´ar’s vanishing is used to prove (3)of (1.2.3), while in positive characteristic this vanishing theorem is known to fail.In fact, [HK12] gives some elementary counter examples to generic vanishing incharacteristic p and shows that generic vanishing results for a generically finiteseperable morphism a : X → A from a smooth variety to an abelian varietyshould be equivalent to the vanishing R i a ∗ ω X = 0 for i >
0. Thus it is clearthat the naive generalization to positive characteristic fails and a new approachis necessary.To establish this new approach, we begin by investigating the fundamentalproperties of the sheaves a ∗ ω X (and R i a ∗ ω X for i ≥ a ∗ ω X is the uppercanonical extension of the lowest piece in the Hodge filtration of the variation ofHodge structures on R k a ∗ C X [Kol86, Theorem 2.6]. In particular, a ∗ ω X is thelowest filtered piece of a filtered D -module on A . The work of Schnell and Popaon generic vanishing clearly illustrates that (in this context) this is the right wayof thinking about a ∗ ω X [PS13]. In positive characteristic Cartier modules, whichare related to D -modules [Lyubeznik97] and are equivalent to ´etale local systemsin the appropriate sense [BB11, BB13], seem to be the correct analog. A Cartiermodule is simply a triple ( M, φ, s ), where M is a coherent sheaf, s > CHRISTOPHER D. HACON AND ZSOLT PATAKFALVI and φ is a homomorphism F s ∗ M → M , for the absolute Frobenius morphism F (if s = 1, we omit it from the notation). One example of Cartier modules is ( ω X , φ ),where φ is the Grothendieck trace of F . In particular, for all integers e ≥ φ e : F e ∗ ω XF e − ∗ ( φ ) / / F e − ∗ ω XF e − ∗ ( φ ) / / . . . φ / / ω X . From equation (1) we obtain many important invariants of a positive charac-teristic variety X , that are the fundamental objects of the current paper. Thefollowing is a short description the most important of these invariants. • For a map a : X → A , applying a ∗ (?) to (1) yields another Cartiermodule ( a ∗ ω X , a ∗ ( φ )). The stable image of a ∗ ( φ e ) (which is the same forall e ≫
0) is denoted by S a ∗ ω X . The natural replacement for a ∗ ω X inpositive characteristic is then either S a ∗ ω X or lim ←− e F e ∗ S a ∗ ω X (we will useboth depending on the context). • Applying H ( X, ? ⊗ L ) for some line bundle L to (1) does not yield aCartier module. However, since H ( X, ω X ⊗ L ) is a finite dimensionalvector space (whenever X is projective over a field), the image of φ e sta-bilizes. This stable image is denoted by S ( X, ω X ⊗ L ). It is a well behavedsubset of all the sections H ( X, ω X ⊗ L ), stable under the Frobenius ac-tion. It is also used to define the Frobenius stable Kodaira dimension κ S ( X ) which has a similar behaviour to the usual Kodaira dimension.For example κ S ( X ) = 0 exactly if dim k S ( X, ω mX ) = 1 for every divisibleenough m ∈ N and if κ S ( X ) = 0 then κ ( X ) = 0. • Further, under mild technical assumptions, we can obtain a Cartier mod-ule starting from a log pair ( X, ∆) instead of a smooth variety X . Thisis somewhat technical, but crucial for our purposes, see (2.2.3).One fundamental reason why it is convenient to replace a ∗ ω X by Ω := lim ←− e F e ∗ S a ∗ ω X is that it satisfies a Kodaira-vanishing type result: H i ( X, Ω ⊗ N ) = 0 for all i > N . The proof is an easy combination of Serre-vanishing and the vanishing of cohomology for inverse limits guaranteed by theML-condition. Using this we show in (3.1.2) that H i ( X, Ω ⊗ ˆ L ∨ ) = 0 for i > H i ( R ˆ S ( D A ( F ))) under the Cartier module action on F = S a ∗ ω X is zero. Theorem 1.3.1. ( cf. (3.1.4) ) Let X be a projective variety over an algebraicallyclosed field k of characteristic p > , a : X → A a morphism to an abelian variety. ENERIC VANISHING AND ORDINARY ABELIAN VARIETIES 5
Then for every i < , lim −→ H i ( R ˆ S ( D A ( F e ∗ S a ∗ ω X ))) = 0 , or equivalently for every integer e ≫ , im( H i ( R ˆ S ( D A ( S a ∗ ω X ))) → H i ( R ˆ S ( D A ( F e ∗ S a ∗ ω X )))) = 0 . We also remark that it is expected that lim ←− F e ∗ R i a ∗ ω X = 0 for i > dim( X/a ( X )).If this is the case, then it is likely that following (1.3.1), one can establish a Frobe-nius stable version of (1.2.2). Further, we remark that we prove a general versionof (1.3.1) in (3.1.4) pertaining to arbitrary Cartier modules. In particular itapplies to the higher direct images R j a ∗ ω X as well.We are unable to prove the analog of (1) of (1.2.3). Theorem (1.3.1) does how-ever imply some weak versions of the more traditional generic vanishing state-ment, see (3.3.1) where it is shown that Corollary 1.3.2. (cf. (3.3.1) ) In the situation of (1.3.1) , set
Ω := lim ←− F e ∗ a ∗ ω X .Then there exists a proper closed subset Z ⊂ ˆ A such that if i > and y ∈ V i (Ω) = { y ∈ ˆ A | h i (Ω ⊗ P y ) = 0 } then p m y ∈ Z for all m ≫ . We also recover a weak version of Simpson’s result (see (3.3.3)).
Theorem 1.3.3. (cf. (3.3.3) ) In the situation of (1.3.1) , if the reduced Pi-card variety of X has no supersingular factors (cf. (2.3) ), then each maxi-mal dimensional component of the closure of the set of points y ∈ ˆ A such that h (Ω ⊗ a ∗ P y ) = 0 is a finite union of torsion translates of abelian subvarieties,where Ω = lim ←− F e ∗ ω X . Similar results for { y ∈ ˆ A | S ( X, ω X ⊗ a ∗ P y ) = 0 } and for the support of Λare obtained in (3.3.2) and (3.3.5). It should be noted that we are unable toprove the analog of the inclusions V i ( ω X ) ⊃ V i +1 ( ω X ) (which holds for m.A.d.projective varieties over C ) nor the result on the reducedness of the loci V i ( ω X ).Even though these results seem to be somewhat technical, it turns out thatthey have several nice applications which mirror the characteristic 0 theory. Inparticular we prove in (4.2.11) and (4.3.1) the previously mentioned (1.1.1). Fur-thermore, we show: Theorem 1.3.4. (cf. (3.2.7) ) If a : X ֒ → A is a closed, smooth subvariety ofgeneral type of an abelian variety, then the smallest abelian subvariety ˆ B ⊆ ˆ A such that the union of finitely many translates of ˆ B contains V ( A, S a ∗ ω X ) isequal to ˆ A . Organization.
In Section 2 we recall the important facts about derivedcategories (2.1), F -singularities (2.2), the Frobenius morphism on abelian vari-eties (2.3), the Fourier-Mukai transform on abelian varieties (2.4), the behaviourof S in families (2.5) and higher direct images of the canonical bundle (2.6). In CHRISTOPHER D. HACON AND ZSOLT PATAKFALVI
Section 3 we first prove our main generic vanishing theorem (3.1) and draw someconsequences (3.2). Then, we prove in subsection (3.3) the statements (1.3.2)and (1.3.3) about cohomology support loci, and finally we give some examples(3.4). In Section 4 we define the Frobenius stable Kodaira dimension (4.1) andthen we prove Theorem (1.1.1) in (4.2) and (4.3).1.5.
Acknowledgments.
The authors would like to thank B. Bhatt and K.Schwede for useful discussions and comments. The first named author was par-tially supported by NSF research grant DMS-0757897 and DMS-1300750 and agrant from the Simons foundation. The second name author would like to thankthe NCTS Mathematics Division (Taipei Office), where part of this article wascompleted during a fruitful visit.2.
Preliminaries
We fix an algebraically closed field k , of characteristic p >
0. All schemes willbe over k unless otherwise stated.2.1. Derived categories.
Let X be a quasi-compact and separated scheme and D ( X ) be its derived category (i.e. the derived category of O X modules). D qc ( X )denotes the full subcategory of D ( X ) consisting of complexes whose cohomologiesare quasi-coherent. For any F ∈ D ( X ), F [ n ] denotes the object obtained byshifting F , n places to the left and H n ( F ) denotes the O X module obtained bytaking the n -th homology of a complex representing F . Recall the following. Theorem 2.1.1 (Projection formula) . Let f : X → Y be a morphism of sepa-rated, quasi-compact schemes, then there is a functorial isomorphism Rf ∗ ( F ) ⊗ O Y G → Rf ∗ ( F ⊗ O X Lf ∗ G ) for any F ∈ D qc ( X ) , G ∈ D qc ( Y ) . (Here ⊗ is taken in the left-derived sense.)Proof. [Neeman96, Proposition 5.3]. (cid:3) If X is a variety of dimension n over a field k and ω · X denotes its dualizing com-plex (which is by definition f ! O k for Hartshorne’s f ! , such that H − dim X ( ω · X ) ∼ = ω X ), then the dualizing functor D X is defined by D X ( F ) = R H om ( F, ω · X ) forany F ∈ D qc ( X ). We have Theorem 2.1.2 (Grothendieck Duality) . Let f : X → Y be a proper morphismof quasi-projective varieties over a field k , then Rf ∗ D X ( F ) = D Y Rf ∗ ( F ) ∀ F ∈ D qc ( X ) . Proof.
For bounded F it is shown in [Hartshorne66, § VII]. The general case is in[Neeman96]. (cid:3)
ENERIC VANISHING AND ORDINARY ABELIAN VARIETIES 7
Definition 2.1.3.
Given a direct system of objects C i ∈ D ( X ) C f / / C f / / . . . hocolim −→ C i is defined by the following triangle L C i / / L C i / / hocolim −→ C i +1 / / , where the first map is the homomorphism given by id − shift, and ”shift” denotesthe map L C i → L C i defined on C i by the composition C i → C i +1 ⊂ L C j .(which is called 1-shift in [Neeman96]). Lemma 2.1.4.
Let C i f i → C i +1 be a direct system in D qc ( X ) . Then homotopycolimits commute with tensor products, pullbacks and pushforwards. In particularwe have (1) hocolim −→ H j ( C i ) = H j (hocolim −→ C i ) , and (2) hocolim −→ R j Γ( C i ) = R j Γ(hocolim −→ C i ) .Proof. See [Neeman96, Lemma 2.8] (with c = O X and T = D qc ( X )) or [Kuznetsov11,2.11]. The fundamental reason is that for any additive functor F from D qc ( X ) tothe category of abelian groups, F (id − shift) is injective. Indeed, let ( c i ) ∈ ⊕ F ( C i ),and let ( d i ) ∈ ⊕ F ( C i ) be the image via F (id − shift). That is, d = c d = c − F ( f )( c ) . . . d i = c i − F ( f i − )( c i − ) . . . Then, we have c = d c = d − F ( f )( d ) c = d − F ( f )( d − F ( f )( d )) . . . In particular, if all d i are zero, then so are all the c i . (cid:3) Definition 2.1.5.
Given an inverse system of objects C i ∈ D qc ( X ) C C f o o . . . ˜ f o o then holim ←− C i is defined by the following triangle [Murfet, Definition 29]holim ←− C i / / Q C i / / Q C i +1 / / . Here the map between products is Q (id − shift). Note also that by productwe mean the ordinary product of chain complexes (which is well-defined on thederived category and is the product in D ( X )), not the product inside D qc ( X ).In particular then holim ←− C i is an object of D ( X ), not of D qc ( X ). CHRISTOPHER D. HACON AND ZSOLT PATAKFALVI
It is easy to check that if C i are coherent sheaves, then hocolim −→ C i = lim −→ C i (see the proof of (2.1.4)). However, holim ←− C i = lim ←− C i in general. For an easyexample, consider C i := O A k , where A = Spec k [ x ], with the maps C i +1 → C i being multiplication by x . Then Q C i → Q C i is not surjective, since (1) ∈ Q O A k is not in the image. Lemma 2.1.6.
Given an inverse system of quasi-coherent sheaves C i C C f o o . . . ˜ f o o satisfying the ML-condition, that is, for every i , im( C j → C i ) is the same for all j ≫ i , we have holim ←− C i = lim ←− C i . Proof.
It is immediate thatlim ←− C i = ker (cid:16)Y C i → Y C i (cid:17) . Hence, to prove the required equality we have to show that Q C i → Q C i issurjective. For that it is enough to prove that Q C i ( U ) → Q C i ( U ) is surjectivefor each affine open set U . However, there the question becomes a question onabelian groups, which is well known (see for example [stacks-project, Tag 07KW,(3)] and [stacks-project, Tag 0594] for the definition of the ML-condition). (cid:3) Further, using the language of [Neeman96, Lemma 2.8] one obtains the follow-ing.
Lemma 2.1.7. If C f / / C f / / . . . is a direct system in D qc ( X ) and D ∈ D qc ( X ) , then RH om (hocolim −→ C i , D ) ∼ = holim ←− RH om ( C i , D ) . Proof.
Apply RH om ( , D ) to the triangle M C i → M C i → hocolim −→ C i +1 → . We obtain the triangle RH om (hocolim −→ C i , D ) → RH om ( M C i , D ) → RH om ( M C i , D ) → +1 . Notice now that(2) RH om ( M C i , D ) ∼ = Y RH om ( C i , D ) . Indeed, for every i there is a natural map RH om ( L C i , D ) → RH om ( C i , D ).This induces a natural map RH om ( L C i , D ) → Q RH om ( C i , D ). To prove thatit is an isomorphism, it is enough to prove that it induces an isomorphism on each ENERIC VANISHING AND ORDINARY ABELIAN VARIETIES 9 cohomology sheaf. By restricting to affine patches we may also replace RH om by R Hom. That is we have to show that Q Hom D ( X ) ( C i , D ) → Hom D ( X ) ( L C i , D )is an isomorphism. This holds because of the universal property of L . Thisconcludes the proof of (2).Then the previous triangle translates to RH om (hocolim −→ C i , D ) → Y RH om ( C i , D ) → Y RH om ( C i , D ) → +1 and the map between the products is just Q (cid:0) id RH om ( C i , D ) −RH om ( f i − , D ) (cid:1) .This shows that RH om (hocolim −→ C i , D ) is indeed the homotopy limit of the inversesystem RH om ( C , D ) RH om ( C , D ) RH om ( f , D ) o o . . . RH om ( f , D ) o o (cid:3) F-singularities.
The title is somewhat misleading. We define here theglobal invariants whose origin lies in the theory of F -singularities. For the generaltheory we refer to [ST] and [S11]. Throughout this subsection the letter F standsfor the absolute Frobenius morphism of the given variety. Definition 2.2.1.
Let X be a smooth, proper variety over k , ∆ ≥ Q -divisor, s > p s − f : X → Y a morphism over k and M a Cartier divisor on X . We define the subsheaf S f ∗ ( σ ( X, ∆) ⊗ O X ( M )) ⊆ f ∗ O X ( M ) to be the intersection: \ e ≥ Image (Tr es F es ∗ f ∗ O X ((1 − p es )( K X + ∆) + p es M ) → f ∗ O X ( M )) , where Tr es is obtained from the Grothendieck trace F es ∗ ω X → ω X of F es bytwisting with O X ( M − K X ), pushing forward by f and applying that F ∗ f ∗ = f ∗ F ∗ .In the special case of Y = Spec k , we use the notation S ( X, σ ( X, ∆) ⊗ O X ( M ))instead of S f ∗ ( σ ( X, ∆) ⊗ O X ( M )).This intersection is a descending intersection, so a priori it needs not stabilize.In this case S f ∗ ( σ ( X, ∆) ⊗ O X ( M )) tends not to be coherent. There are severalcases when the intersection stabilizes, for example if M − K X − ∆ is ample (cf.[HX13, 2.15]). Lemma 2.2.2.
With the above notation, there is a natural inclusion S ( X, σ ( X, ∆) ⊗O X ( M )) ⊆ H ( Y, S f ∗ ( σ ( X, ∆) ⊗ O X ( M ))) .Proof. Denote F e := F es ∗ f ∗ O X ((1 − p es )( K X + ∆) + p es M ) . Then, S ( X, σ ( X, ∆) ⊗ O X ( M )) = \ e ≥ im( H ( Y, F e ) → H ( Y, F )) , while H ( Y, S f ∗ ( σ ( X, ∆) ⊗ O X ( M ))) = H Y, \ e ≥ im( F e → F ) ! . The inclusion then follows from the following computation. \ e ≥ im( H ( Y, F e ) → H ( Y, F )) ⊆ \ e ≥ H ( Y, im( F e → F ))= H Y, \ e ≥ im( F e → F ) ! . (cid:3) Lemma 2.2.3. [Pat13, Lemma 2.6]
Let X be a smooth variety and D ∈ | mK X | for some integer m > coprime to p . Assume further that f : X → Y is a propermorphism over k . Define s > to be the smallest integer such that m | ( p s − and ∆ := m − m D . Then the chain (3) · · · → f ∗ F ( e +1) s ∗ O X ( mp ( e +1) s K X + (1 − p ( e +1) s )( K X + ∆)) →→ f ∗ F es ∗ O X ( mp es K X + (1 − p es )( K X + ∆)) → . . . is isomorphic to (4) · · · → f ∗ F ( e +1) s ∗ O X ( mK X ) f ∗ F es ∗ ( α ) −−−−−→ f ∗ F es ∗ O X ( mK X ) f ∗ F ( e − s ∗ ( α ) −−−−−−−→ f ∗ F ( e − s ∗ O X ( mK X ) → . . . , where α is the usual homomorphism induced by the Grothendieck trace of Frobe-nius F s ∗ O X ( mK X ) ∼ = F s ∗ O X ( mp s K X + (1 − p s )( K X + ∆)) → O X ( mK X ) . In particular, the intersection in the definition of Ω := S f ∗ ( σ ( X, ∆) ⊗O X ( mK X )) stabilizes by [BS12, Proposition 8.1.4] and agrees with the image of f ∗ F ( e +1) s ∗ O X ( mK X ) → f ∗ O X ( mK X ) for e ≫ . Furthermore, these then form a chain of surjective maps · · · ։ F s ∗ Ω ։ F s ∗ Ω ։ Ω . Lemma 2.2.4. If a : X → A is a proper morphism from a smooth variety X toa scheme A over k , then S ( X, ω X ) = S ( A, S a ∗ ω X ) .Proof. Suppose that f ∈ S ( A, S a ∗ ω X ), then there are elements f e ∈ H ( A, F e ∗ S a ∗ ω X ) ⊂ H ( A, F e ∗ a ∗ ω X ) ∼ = H ( X, F e ∗ ω X ) ENERIC VANISHING AND ORDINARY ABELIAN VARIETIES 11 such that H ( A, a ∗ (tr F e ))( f e ) = f for all e ≥
0. We write ˜ f and ˜ f e for thecorresponding elements in H ( X, ω X ) and H ( X, F e ∗ ω X ). Since the followingdiagram commutes H ( A, F e ∗ S a ∗ ω X ) / / (cid:127) _ (cid:15) (cid:15) H ( A, S a ∗ ω X ) (cid:127) _ (cid:15) (cid:15) H ( A, F e ∗ a ∗ ω X ) / / H ( A, a ∗ ω X ) H ( X, F e ∗ ω X ) / / H ( X, ω X ) , it follows that ˜ f = H ( X, tr F e )( ˜ f e ) so that ˜ f ∈ S ( X, ω X ). Thus S ( X, ω X ) ⊃ S ( A, S a ∗ ω X ).For the reverse inclusion, note that if φ : F ∗ ω X → ω X is the Grothendiecktrace, then ( a ∗ ω X , a ∗ ( φ )) is a Cartier module. Hence, by [BS12, Proposition8.1.4], there is an e , such that F e ∗ a ∗ ω X → S a ∗ ω X is surjective. Let then e ′ ≥ H ( X, F e + e ′ ∗ ω X ) = H ( A, a ∗ F e + e ′ ∗ ω X ) H ( X, tr F e + e ′ ) ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ H ( A,F e ′∗ a ∗ (tr F e )) / / H ( A, F e ′ ∗ S a ∗ ω X ) H ( A,a ∗ ( tr F e ′ )) (cid:15) (cid:15) H ( A, S a ∗ ω X ) (cid:127) _ (cid:15) (cid:15) H ( A, a ∗ ω X ) = H ( X, ω X ) (cid:3) The Frobenius morphism on abelian varieties.
Throughout this paper A will denote an abelian variety of dimension g over k , ˆ A = Pic ( A ) the dualabelian variety and L the normalized Poincar´e line bundle on A × ˆ A . Further V A and V ˆ A are the Verschiebung isogenies [MvdG, 5.18], where we drop the subindexwhenever it is clear from the context.Given a scheme X over k with structure morphism µ : X → Spec k , we denoteby X ′ the twisted version of X , i.e., the scheme over k which is identical to X asan abstract scheme but its k -structure is F k ◦ µ . If instead of F k we compose with F mk then we write X ( m ) , where m can be negative as well. The main reason forintroducing X ′ , and X ( m ) in general, is that the absolute Frobenius morphism F : X → X is not a k -morphism. To treat it as a k -morphism one has to regardit as a morphism X ′ → X . For many purposes (e.g., the content of Subsection2.2), this is not necessary and taking the domain of the Frobenius to be X as wellis more convenient (e.g., for defining iterations of a Frobenius action). However, when the k -structure is important, e.g., when one takes the product of X withanother k -variety (which we will do frequently), then it is important to treat thedomain of the Frobenius as X ′ . Further, our notions will be based on dimensionsof k -vector spaces, which is invariant, by the perfectness assumption on k , undertwisting the k -structure by the Frobenius morphism of k . Hence we will freelychange between the two point of views, according to which is more adequate,sometimes leaving a few details to the reader. Lemma 2.3.1.
Given an abelian variety A over k , the dual of A ′ is isomorphicto ( ˆ A ) ′ (hence we denote both by ˆ A ′ ). Further the Verschiebung V : ˆ A → ˆ A ′ isthe dual of the Frobenius F : A ′ → A . The proof is an straight forward application of the Seesaw principle.
Proposition 2.3.2.
Let A be an abelian variety over k of dimension n . Thefollowing are equivalent. (1) A is ordinary in the sense of [BK86, 7.2] , (2) there are p g p -torsion points, (3) V is ´etale, (4) the Frobenius action H n ( A, O A ) → H n ( A, O A ) is bijective, (5) the Frobenius action H i ( A, O A ) → H i ( A, O A ) is bijective for ≤ i ≤ n , (6) A is globally F-split, (7) S ( A, ω A ) = 0 . Note that apart from point (2), the rest is equivalent also over non algebraicallyclosed perfect fields, because they are properties that are invariant under base-field extension.
Proof. (1) and (2) are equivalent by [BK86, 7.4].(2) and (3) are equivalent because as V : ˆ A → ˆ A is an isogeny, V is the quotientby the scheme theoretic inverse image V − (0). Hence V is ´etale if and only if V − (0) is reduced. However V − (0) has always length p g (because deg V = p g ),and so V − (0) is reduced if and only if it contains p g points. Since the points of V − (0) are exactly the p -torsion points, we are done.(2) and (4) are equivalent by [MS11, 5.4].(4) and (5) are equivalent by [MS11, 5.4].(4) and (7) are equivalent because the morphism O A → F ∗ O A , which inducesthe map of (4) is dual to the Grothendieck trace. So, dualizing the map of (4)one obtains H ( A, F ∗ ω A ) → H ( A, ω A ).(6) and (7) are well known to be equivalent. (cid:3) If any of the above equivalent conditions holds, we say that A is ordinary . Wehave the following easy lemma. Lemma 2.3.3.
Let ϕ : A → B be an isogeny between abelian varieties of di-mension n , then A is ordinary if and only if B is ordinary. In particular A isordinary if and only if ˆ A is ordinary. ENERIC VANISHING AND ORDINARY ABELIAN VARIETIES 13
Proof.
This is immediate by (2) of (2.3.2) and [MvdG, 5.22]. For the addendum,note that any abelian variety is isogenous to its dual abelian variety. (cid:3)
Lemma 2.3.4.
Let ϕ : A → B be a surjective morphism of abelian varieties. If A is ordinary, then so is B .Proof. Let d = dim B . By (2.3.3) and [Mumford70, p. 173] we may assumethat A → B is a projection onto a factor. By (5) of (2.3.2), F ∗ A : H d ( A, O A ) → H d ( A, O A ) is bijective. Since ϕ : H d ( B, O B ) → H d ( A, O A ) is injective and F B ◦ ϕ = ϕ ◦ F A , it follows that F ∗ B : H d ( B, O B ) → H d ( B, O B ) is bijective andhence that B is ordinary. (cid:3) Following [PR03] we say that an abelian variety A endowed with an isogeny ϕ : A → A is pure of positive weight if there exists r, s > ϕ s = F p r forsome model of A over F p r . In particular ϕ is purely inseperable. If A is definedover a finite field, then A is supersingular if and only if it is pure of positive weightfor the isogeny given by multiplication by p , in general A is supersingular if andonly if it is isogenous to a supersingular abelian variety defined over a finite field.Further, as in [PR03], A has no supersingular factor if there does not exists anontrivial homomorphism to a supersingular abelian variety.Since for an ordinary abelian variety [ p ] = V ◦ F is never purely inseparableby (2.3.2), an ordinary abelian variety is never pure of positive weight for theisogeny given by multipliation by p . So, we have the following. Lemma 2.3.5. If A is an ordinary abelian variety, then A has no supersingularfactors.Proof. Immediate from (2.3.4). (cid:3)
Theorem 2.3.6.
Let A be an abelian variety over k and X ⊂ A a reduced closedsubscheme satisfying p ( X ) = X (resp. p ( X ) ⊂ X ). If A has no supersingularfactors, then X is completely linear i.e. a finite union of torsion translates of subabelian varieties (resp. all maximal dimensional irreducible components of X arecompletely linear).Proof. See [PR03, 2.2] and the proof of [PR03, 4.1]. (cid:3)
Abelian varieties and the Fourier-Mukai functor.
The Fourier-Mukaitransforms R ˆ S : D ( A ) → D ( ˆ A ) and RS : D ( ˆ A ) → D ( A ) are defined by R ˆ S (?) = Rp ˆ A, ∗ ( Lp ∗ A ? ⊗ L ) , RS (?) = Rp A, ∗ ( Lp ∗ ˆ A ? ⊗ L ) . Note that p ∗ A and p ∗ ˆ A are exact so, sometimes L is omitted in front of them. By[Mukai81], it is known that: Theorem 2.4.1.
The following equalities hold on D qc ( A ) and D qc ( ˆ A ) . RS ◦ R ˆ S = ( − A ) ∗ [ − g ] , and R ˆ S ◦ RS = ( − ˆ A ) ∗ [ − g ] , where [ − g ] denotes the shift by g places to the right and − A is the inverse on A . The following two properties are proven in [Mukai81, 3.1 and 3.8].
Lemma 2.4.2. (Exchange of translation and ⊗ Pic .) Let x ∈ ˆ A and P x = L| A × x ,then the following equalities hold on D qc ( A ) . RS ◦ T ∗ x ∼ = ( ⊗ P − x ) ◦ RS.
Lemma 2.4.3.
We have D A ◦ RS ∼ = (( − A ) ∗ ◦ RS ◦ D ˆ A )[ g ] on D qc ( ˆ A ) . Note that in the statement of the following lemma we consider the functors onthe whole D ( A ) not only the subcategory D qc ( ˆ A ). Lemma 2.4.4.
For every x ∈ A , T ∗ x ◦ D A ∼ = D A ◦ T ∗ x as functors on D ( A ) .Proof. By the proof of [Hartshorne77, Proposition III.2.2] one can choose aninjective resolution I of ω • A , for which T ∗ x I ∼ = I (as complexes, not just a quasi-isomorphism). Then one can compute T ∗ x ◦ D A ( D ) as T ∗ x ( H om • ( D , I )) and D A ◦ T ∗ x ( D ) as H om • ( T ∗ x D , I ). Note here we have used that T x is an automorphism,hence T ∗ x is exact. Further note that T ∗ x ( H om • ( D , I )) ∼ = H om • ( T ∗ x D , T ∗ x I ) (bychecking it on every open set), hence it suffices to prove that H om • ( T ∗ x D , T ∗ x I )and H om • ( T ∗ x D , I ) are quasi-isomorphic. However, they are even isomorphic bythe choice of I . (cid:3) The following result on the exchange between direct and inverse images iscontained in [Mukai81, 3.4].
Lemma 2.4.5.
Let φ : A → B be an isogeny of abelian varieties and ˆ φ : ˆ B → ˆ A the dual isogeny, then the following equalities hold on D qc ( B ) and D qc ( A ) . φ ∗ ◦ RS B ∼ = RS A ◦ ˆ φ ∗ φ ∗ ◦ RS A ∼ = RS B ◦ ˆ φ ∗ . Proof.
See [Mukai81, 3.4]. (cid:3)
Lemma 2.4.6.
Let Λ e → Λ e +1 be a direct system in D qc ( A ) . Then hocolim −→ R ˆ S (Λ e ) = R ˆ S (hocolim −→ Λ e ) .Proof. We have R ˆ S (hocolim −→ Λ e ) = Rp ˆ A, ∗ ( Lp ∗ A (hocolim −→ Λ e ) ⊗ L ) =hocolim −→ Rp ˆ A, ∗ ( Lp ∗ A Λ e ⊗ L ) = hocolim −→ R ˆ S (Λ e ) , where the first and last equalities follow from the definition of R ˆ S and the middleequality follows by (2.1.4). (cid:3) ENERIC VANISHING AND ORDINARY ABELIAN VARIETIES 15 S ( ω X ) in families. By the following theorem from [Pat13, Theorem 3.3],the Frobenius stable subspace S ⊆ H behaves well in families. We cite hereonly the special case used in this paper. For the general case please see [Pat13]. Theorem 2.5.1. [Pat13, Theorem 3.3]
Let f : X → Y be a proper, surjective,generically smooth morphism of smooth varieties over k . Then, there is a non-empty Zariski open set W of Y such that S ( F, ω F ) has the same dimension forevery geometric fiber F over W . Further, the rank of S f ∗ ω X is at least as bigas this general value. Higher direct images of ω X .Proposition 2.6.1. Let f : X → A be a generically finite dominant mor-phism of projective varieties. Assume that X is smooth of dimension n . Then codim R i f ∗ ω X ≥ i + 2 for all i > .Proof. We proceed by induction on the dimension. Let H be a very generalsufficiently ample divisor. Pushing forward the short exact sequence0 → ω X → ω X ( H ) → ω H → , one sees that it is enough to prove that codim R f ∗ ω X ≥
3. This can be checkedby localizing at a codimension 2 point, in which case it is a consequence of therelative Kawamata-Viehweg vanishing (which holds for two dimensional excellentschemes see [KK94, 2.2.5]). (cid:3) Generic Vanishing
Proof of Theorem 1.3.1.
Recall that in our notation k is an algebraicallyclosed field of characteristic p > A an abelian variety defined over k . Theorem 3.1.1.
Let ψ e : Ω e +1 → Ω e be an inverse system of coherent sheaveson an abelian variety A such that for any sufficiently ample line bundle L on ˆ A and any e ≫ , H i ( A, Ω e ⊗ ˆ L ∨ ) = 0 for all i > . Then, the complex Λ := hocolim −→ R ˆ S ( D A (Ω e )) is a quasi-coherent sheaf in degree , i.e., Λ = H (Λ) . Furthermore, if there isan integer r > , such that the image Ω e ′ → Ω e is the same for every e ′ ≥ e + r ,then Ω := lim ←− Ω e = (( − A ) ∗ D A RS (Λ))[ − g ] . Proof.
The object Ω e lives in degree 0, hence D A (Ω e ) lives in degrees [ − g, . . . , H j ( D A (Ω e ))) ≥ g + j. We then have that Λ e := R ˆ S ( D A (Ω e )) lives in degrees [ − g, . . . , −→ Λ e . By (2.1.4), Λ also lives in degrees [ − g, . . . ,
0] andit has quasi-coherent cohomologies.
To show the statement about Λ we must show that Λ lives in cohomologicaldegree
0, i.e., that H j (Λ) = 0 for j ∈ [ − g, . . . , − j ∈ [ − g, . . . ,
0] be the smallest integer such that H j (Λ) = 0 andassume that j ≤ −
1. We have that H j (Λ) = hocolim −→ H j (Λ e ) (cf. (2.1.4)) and sowe may fix e > H j (Λ e ) → H j (Λ) is non-zero. We twistby a sufficiently ample line bundle L so that H j (Λ e ) ⊗ L is globally generatedand hence the image of R Γ( H j (Λ e ) ⊗ L ) → R Γ( H j (Λ) ⊗ L )is non-zero. Let E i,l = R i Γ( H l (Λ) ⊗ L )abutting to R i + l Γ(Λ ⊗ L ). By our choice of j we have that E i,l = 0 for l < j andhence that(5) R j Γ(Λ ⊗ L ) ∼ = R Γ( H j (Λ) ⊗ L ) = E ,j = 0 . On the other hand, following the beginning of the proof of [Hacon04, 1.2], wehave that D k ( R Γ(Ω e ⊗ ˆ L ∨ )) ∼ = G . D . R Γ( D A (Ω e ⊗ ˆ L ∨ )) ∼ = R Γ( D A (Ω e ) ⊗ ˆ L ) ∼ = R Γ( D A (Ω e ) ⊗ p A, ∗ ( L ⊗ p ∗ ˆ A L )) ∼ = P . F . R Γ( Lp ∗ A D A (Ω e ) ⊗ L ⊗ p ∗ ˆ A L )) ∼ = P . F . R Γ( R ˆ SD A (Ω e ) ⊗ L ) = R Γ(Λ e ⊗ L ) . Since, by assumption, R l Γ(Ω e ⊗ ˆ L ∨ ) = 0 for any e ≫ l >
0, it follows that R − l Γ(Λ e ⊗ L ) = 0 for any e ≫ R j Γ(Λ e ⊗ L ) = 0 for any e ≫ R j Γ(Λ ⊗ L ) = lim −→ R j Γ(Λ e ⊗ L ) = 0(cf. (2.1.4)). This contradicts (5) and hence concludes the first part of ourstatement.The second part is shown by the following stream of isomorphisms. D A RS (hocolim −→ Λ e ) ∼ = D A (hocolim −→ RSR ˆ SD A (Ω e )) | {z } (2.4.6) ∼ = holim ←− D A (( − A ) ∗ D A (Ω e )[ − g ]) | {z } (2.1.7)+(2.4.1) ∼ = (holim ←− D A ( − A ) ∗ D A (Ω e ))[ g ] | {z } ( − A ) ∗ = ( − A ) ∗ , because − A ◦ − A = Id A ∼ = ( − A ) ∗ (holim ←− D A D A (Ω e ))[ g ] | {z } ( − A ) ∗ D A = D A ( − A ) ∗ by G.D. ∼ = ( − A ) ∗ holim ←− Ω e [ g ] | {z } D A D A ( F ) = F for F coherent ∼ = ( − A ) ∗ Ω[ g ] | {z } (2.1.6) . (cid:3) ENERIC VANISHING AND ORDINARY ABELIAN VARIETIES 17
Choose a coherent sheaf Ω on A and an integer s >
0, such that there is ahomomorphism α : Ω := F s ∗ Ω → Ω (i.e., (Ω , α, s ) is a Cartier module, c.f.,Subsection 1.3). Then this induces for every integer e ≥ F es ∗ ( α ) : Ω e +1 := F ( e +1) s ∗ Ω → Ω e := F es ∗ Ω . An example of such setup is that of (2.2.3), by setting Y := A and definingΩ := S f ∗ ( σ ( X, ∆) ⊗ O X ( mK X )). Notice that for any Cartier module (Ω , α, s ),there exists an integer e and a coherent subsheaf Ω ′ ⊂ Ω ([Gabber04, Lemma13.1] [BS12, Proposition 8.1.4]) such thatΩ ′ = Im (Ω e → Ω ) . We then have that Ω ∼ = lim ←− F es ∗ Ω ′ . Thus replacing Ω by Ω ′ we may assume thatthe homomorphisms Ω e +1 → Ω e are surjective. Lemma 3.1.2.
With notation as above, let L be an ample line bundle on ˆ A and ˆ L = RS ( L ) = R S ( L ) be its Fourier Mukai transform. If Ω e = F es ∗ Ω , then H i ( A, Ω e ⊗ ˆ L ∨ ⊗ P ) = 0 for e ≫ , any i > and P ∈ ˆ A .Proof. Recall that φ ∗ L ( ˆ L ∨ ) ∼ = L ⊕ h ( L ) . Note that H i ( A, Ω e ⊗ ˆ L ∨ ⊗ P ) ∼ = H i ( A, F esA, ∗ Ω ⊗ ˆ L ∨ ⊗ P ) ∼ = proj . formula H i ( A, Ω ⊗ F es, ∗ A ( ˆ L ∨ ⊗ P )) ∼ = H i ( A, Ω ⊗ F es, ∗ A ( ˆ L ∨ ) ⊗ P p es )and (by Cohomology and Base Change) the required vanishing is equivalent toshowing that R ˆ S (Ω ⊗ F es, ∗ A ( ˆ L ∨ )) is a sheaf (in degree 0) for every e ≫
0. (Wehave used the fact that ˆ A is p -divisible, so for any Q ∈ ˆ A there exists a P ∈ ˆ A with Q = P p es .) This is equivalent to showingˆ φ L, ∗ R ˆ S (Ω ⊗ F es, ∗ A ( ˆ L ∨ )) = R ˆ S ( φ ∗ L (Ω ⊗ F es, ∗ A ( ˆ L ∨ )))is a sheaf for every e ≫ H i ( ˆ A, φ ∗ L Ω ⊗ φ ∗ L F es, ∗ A ( ˆ L ∨ ) ⊗ P ) = H i ( ˆ A, φ ∗ L (Ω ⊗ F es, ∗ A ( ˆ L ∨ )) ⊗ P ) = 0for e ≫ i > P ∈ ˆ A (where e is independent of P ). Since F esA ◦ φ L = φ L ◦ F es ˆ A , we have φ ∗ L F es, ∗ A ( ˆ L ∨ ) = F es, ∗ ˆ A φ ∗ L ( ˆ L ∨ ) = F es, ∗ ˆ A M h ( L ) L = M h ( L ) L p es , and so the last vanishing is immediate (for e ≫
0) from Serre-Fujita vanishing[Fujita82]. (cid:3)
From now on we will adopt the following notation.
Notation 3.1.3.
Let Ω be a coherent sheaf on A and s > α : Ω := F s ∗ Ω → Ω (i.e., a Cartier module). Unlessotherwise specified, Ω is arbitrary. We also fix the following notation throughoutthe artice: Ω e := F es ∗ Ω , Ω := lim ←− Ω e , Λ e := R ˆ S ( D A (Ω e )), Λ := lim −→ Λ e . Corollary 3.1.4.
With the above notation, Λ is a quasi-coherent sheaf and Ω =( − A ) ∗ D A RS (Λ)[ − g ] .Proof. By (3.1.2) and (3.1.1). (cid:3)
Proof of (1.3.1) . Choose Ω := S a ∗ ω X . Then by (3.1.4),0 = H i (Λ) = H i (hocolim −→ R ˆ S ( D A ( F e ∗ S a ∗ ω X ))) = lim −→ H i ( R ˆ S ( D A ( F e ∗ S a ∗ ω X ))) | {z } (2.1.4) . (cid:3) Consequences of Theorem 3.1.1.
First, we present a corollary that isnot a consequence of (3.1.1), but it is used frequently from here on. Then welist technical statements, most of which are used in Section 4, except (3.2.4) and(3.2.5) that are used already in (3.2.7). Note that the notations of (3.1.3) areassumed from here.
Corollary 3.2.1.
For every closed point y ∈ ˆ A , we have Λ ⊗ k ( y ) ∼ = lim −→ H ( A, Ω e ⊗ P ∨ y ) ∨ ∼ = lim −→ H ( A, Ω ⊗ P − p e y ) ∨ , and for every closed point y ∈ ˆ A and integer e ≥ , H (Λ e ) ⊗ k ( y ) ∼ = H ( A, Ω e ⊗ P ∨ y ) ∨ ∼ = H ( A, Ω ⊗ P − p e y ) ∨ . Proof.
Note first that Λ e is supported in cohomological degrees [ − g, . . . ,
0] asexplained in the proof of (3.1.1). Hence, by cohomology and base change, andfor any y ∈ ˆ A we have H (Λ e ) ⊗ k ( y ) = H ( R ˆ SD A (Ω e )) ⊗ k ( y ) = R Γ( D A (Ω e ) ⊗ P y ) = H ( A, Ω e ⊗ P ∨ y ) ∨ . Since Λ = lim −→ H ( R ˆ SD A (Ω e )), it follows by (2.1.4) that Λ ⊗ k ( y ) ∼ = lim −→ H ( A, Ω e ⊗ P ∨ y ) ∨ . (cid:3) Corollary 3.2.2.
Suppose that H ( A, Ω ⊗ P y ) = 0 for all y ∈ ˆ A , then Λ = 0 and
Ω = 0 .Proof.
By (3.2.1), Λ e = 0 for every e . Hence Λ = 0 and then Ω = ( − A ) ∗ D A RS (0)[ − g ] =0 by (3.1.1). (cid:3) Recall that traditionally cohomology and base change is stated for cohomology of coherentsheaves, however it also applies for hypercohomologies of bounded complexes cf. [EGA III, 7.7,7.7.4, 7.7.12(ii)] and the remark on [PP11, 3.6].
ENERIC VANISHING AND ORDINARY ABELIAN VARIETIES 19
Recall that a unipotent vector bundle is a given by finitely many successiveextensions of line bundles P ∈ ˆ A or equivalently the Fourier Mukai transform ofan Artinian module of finite rank on ˆ A . Corollary 3.2.3. If Λ has a non-zero direct factor which is a direct limit ofArtinian coherent sheaves, then RS (Λ) has a non-zero direct factor of the form lim −→ V e where V e are unipotent vector bundles and the maps V e → V e +1 are injectivewith cokernel being a unipotent vector bundle as well. In particular, Supp Ω =Supp D A ( RS (Λ))[ − g ] = A (recall that D A ( RS (Λ))[ − g ] is a sheaf by (3.1.4) ).Proof. By assumption, Λ = B ⊕ lim −→ G e where G e are Artinian sheaves, lim −→ G e = 0and B is some quasi-coherent sheaf. Then RS (lim −→ G e ) is a direct factor of RS (Λ).Thus, we may assume that Λ = lim −→ G e . Replacing G e by the image of G e → Λ,we may further assume that the maps G e → G e +1 are injective. Now, since G e isArtinian, H i ( ˆ A ⊗ k k ( x ) , G e ⊗ P x ) = 0 for every i > x ∈ A . Therefore, bycohomology and base-change V e := RS ( G e ) is a vector bundle. Further there isan exact sequence0 / / V e / / V e +1 / / W e := RS (coker( G e → G e +1 )) / / V e and W e are unipotent because Artinian coherent sheaves have afiltration by skyscraper sheaves of length one. Then by (2.4.6)(6) RS (Λ) ∼ = RS (lim −→ G e ) ∼ = hocolim −→ RS ( G e ) = lim −→ V e , and furthermore since the homomorphism V e → V e +1 are injective, RS (Λ) = 0.By (3.2.4) and (3.1.4), to prove the support statement it is enough to showthat Supp D A ( RS (Λ))[ − g ] = A . So, by (6) D A ( RS (Λ))[ − g ] = D A (lim −→ V e )[ − g ] =lim ←− D A ( V e )[ − g ] (note that since the V e are unipotent vector bundles, D A ( V e )[ − g ]are sheaves), where the D A ( V e )[ − g ] fit into exact sequences of unipotent vectorbundles 0 / / D A ( W e )[ − g ] / / D A ( V e +1 )[ − g ] / / D A ( V e )[ − g ] / / . Therefore D A ( RS (Λ))[ − g ] is an inverse limit of unipotent vector bundles withsurjective maps between them. Thus, Supp D A ( RS (Λ))[ − g ] = A . (cid:3) Lemma 3.2.4. If α : F s ∗ Ω → Ω is surjective, then Supp Ω = Supp Ω .Proof. Let P ∈ A . There are two cases: • If P / ∈ Supp Ω , then for every open set U ⊆ A \ Supp Ω , Ω ( U ) = 0.Therefore also Ω e ( U ) = 0. Hence (lim ←− Ω e )( U ) = lim ←− (Ω e ( U )) = 0. Inparticular, Ω P = 0, and therefore P / ∈ Supp Ω. • If P ∈ Supp Ω , then choose an affine open set U ∋ P and an element s ∈ Ω ( U ), such that its image in (Ω ) P is not zero. Since U is affine,there is a chain of elements s e ∈ Ω e ( U ) such that s e maps onto s e − foreach e >
0. Therefore ( s e | e ≥ ∈ lim ←− Ω e ( U ) = (lim ←− Ω e )( U ) defines an element the restriction of which to any V ⊆ U is not zero, because( s e | e ≥ | V = ( s e | V | e ≥
0) and s | V = 0 by the choice of s . Therefore,this defines a non-zero element of Ω P , which shows that P ∈ Supp Ω. (cid:3)
Corollary 3.2.5.
Assume that F s ∗ Ω → Ω is surjective. Let ˆ B ⊂ ˆ A be anabelian subvariety such that V (Ω ) = { P ∈ ˆ A | h (Ω ⊗ P ) = 0 } is contained in finitely many translates of ˆ B . Then T ∗ x Ω ∼ = Ω for every x ∈ [ ˆ A/ ˆ B .In particular Supp Ω (which is a closed subvariety by (3.2.4) ) is fibered by theprojection A → B (i.e., Supp Ω is a union of fibers of A → B ).Proof. By (3.2.1), the sheaf H (Λ ) is supported on V (Ω ). Let ˆ K = ˆ A/ ˆ B ,then as V (Ω ) is contained in finitely many fibers of π : ˆ A → ˆ K , it followsthat H (Λ ) ⊗ π ∗ P ∼ = H (Λ ) for all P ∈ Pic ( ˆ K ) = K ⊂ Pic ( ˆ A ) = A . Sinceˆ F es : ˆ A → ˆ A is an isogeny, for any P ∈ Pic ( ˆ A ) and any e > Q ∈ Pic ( ˆ A ) such that ˆ F es, ∗ Q ∼ = P . If moreover P ∈ π ∗ Pic ( ˆ K ), then we mayassume that Q ∈ π ∗ Pic ( ˆ K ). By (2.4.5), it follows that H (Λ e ) ⊗ P ∼ = ( ˆ F es, ∗ H (Λ )) ⊗ P ∼ = ˆ F es, ∗ ( H (Λ ) ⊗ Q ) ∼ = ˆ F es, ∗ H (Λ ) ∼ = H (Λ e ) . But then(7) Λ ⊗ P = H (Λ) ⊗ P = lim −→ H (Λ e ) ⊗ P ∼ = Λand so T ∗ x Ω ∼ = T ∗ x (( − A ) ∗ ( D A RS (Λ))[ − g ]) ∼ = ∼ = (( − A ) ∗ T ∗− x D A RS (Λ))[ − g ] ∼ = (( − A ) ∗ D A T ∗− x RS (Λ))[ − g ] | {z } (2.4.4) ∼ = (( − A ) ∗ D A RS (Λ ⊗ P x ))[ − g ] | {z } (2.4.2) ∼ = (( − A ) ∗ D A RS (Λ))[ − g ] | {z } (7) ∼ = Ω . Therefore Supp Ω is invariant under T x for every x ∈ K , which concludes ourproof. (cid:3) Remark 3.2.6.
The image of Supp Ω in B can have positive dimension, even ifone takes ˆ B to be the smallest abelian subvariety as above. For example if onetakes the embedding a : C → A of a curve of genus at least two into its Jacobian,and Ω := S a ∗ ω C . Then Ω = a ∗ ω C and Supp Λ = ˆ A . Thus, ˆ A = ˆ B , and hence A = B . So, the image of Supp Ω = Supp Ω in B is isomorphic to C . Theorem 3.2.7. If a : X ֒ → A is a closed, smooth subvariety of general typeof an abelian variety, then the smallest abelian subvariety ˆ B ⊆ ˆ A such that theunion of finitely many translates of ˆ B contains V ( A, S a ∗ ω X ) is equal to ˆ A . ENERIC VANISHING AND ORDINARY ABELIAN VARIETIES 21
Proof.
Assume that ˆ B ( ˆ A . Set Ω = S a ∗ ω X (= a ∗ ω X , because X is smooth).By (3.2.5) and (3.2.4), there is a fibration h : X → Y such that every fiber is apositive dimensional abelian variety. Hence ω X | G ∼ = O G for the general fiber G of h . However, this contradicts the fact that ω X is big. (cid:3) Remark 3.2.8.
It is easy to generalize (3.2.7) to the case when X is mildlysingular in an adequate sense. We leave this to the interested reader since we willnot need this in what follows.3.3. Frobenius stable cohomology support loci (proof of (1.3.2) and (1.3.3) ). We do not have an optimal definition of Frobenius stable cohomologysupport locus (see below for the different variants). So, we present separate state-ments for a few different possible candidates. Recall that (3.1.3) is assumed forthis section and the rest of the article.
Corollary 3.3.1.
There exists a proper closed subset Z ⊂ ˆ A such that if i > and p e y Z for infinitely many e ≥ , then lim −→ H i ( A k ( y ) , Ω e ⊗ P ∨ y ) ∨ = 0 orequivalently lim ←− H i ( A k ( y ) , Ω e ⊗ P ∨ y ) = 0 . Let W i = { y ∈ ˆ A | lim ←− H i ( A k ( y ) , Ω e ⊗ P ∨ y ) = 0 } , then W i ⊂ Z ′ = S (cid:16) [ p e ˆ A ] − ( Z ) (cid:17) red .Proof. Let Z be the proper closed subset where H i (Λ ) is not locally free for any0 ≤ i ≤ g . Note that Λ e = V e, ∗ Λ , where V is the Verschiebung which takes P y to P p e y = P p e y (cf. (2.4.5) and (2.3.1)). In particular, if y is as above, that is, p e y Z for infinitely many e ≥
0, then H i (Λ e ) is locally free at y for infinitelymany e ≥ i . In particular, by cohomology and base change (similarlyto the proof of (3.2.1)), for infinitely many e ≥ H − i (Λ e ) ⊗ k ( y ) ∼ = H i ( A k ( y ) , Ω e ⊗ P ∨ y ) ∨ . Since lim −→ H − i (Λ e ) = 0 by (3.1.1), it follows that lim −→ H i ( A k ( y ) , Ω e ⊗ P ∨ y ) ∨ = 0.Note that lim ←− H i ( A k ( y ) , Ω e ⊗ P ∨ y ) = D k ( y ) (lim −→ H i ( A k ( y ) , Ω e ⊗ P ∨ y ) ∨ ). (cid:3) Proposition 3.3.2.
Let X be a smooth, projective variety over k , a : X → A the Albanese morphism of X and define V S := { P ∈ ˆ A | S ( X, ω X ⊗ a ∗ P ) = 0 } .Then: (1) V S ⊂ V S is the complement of countably many locally closed subsets. (2) Whenever P ∈ V S , we also have P p ∈ V S . If moreover A has no super-singular factors, then each maximal dimensional irreducible component ofthe closure of V S is a torsion translate of an abelian subvariety of ˆ A .Proof. We first prove point (2). Define S e, := im( H ( X, F e ∗ ( ω X ⊗ a ∗ P p e )) → H ( X, F ∗ ( ω X ⊗ a ∗ P p ))) . Then, H ( X, F ∗ ω X ⊗ a ∗ P p ) = S , ⊇ S , ⊇ S , ⊇ . . . . Suppose that P ∈ V S . Since Tr( S e, ) = S ( X, ω X ⊗ a ∗ P ) = 0 for every e ≫ S e, = 0 for every integer e >
0. Since pushing forward via F induces isomor-phisms H ( X, F e − ∗ ( ω X ⊗ a ∗ ( P p ) p e − )) ∼ = H ( X, F e ∗ ( ω X ⊗ a ∗ P p e )), it follows that S e, ∼ = im( H ( X, F e − ∗ ( ω X ⊗ ( a ∗ P p ) p e − )) → H ( X, ω X ⊗ a ∗ P p )) . Thus S ( X, ω X ⊗ a ∗ P p ) = 0. By (2.3.6), if A has no supersingular factors, onesees that maximal dimensional irreducible component of V S is a finite union oftorsion translates of abelian subvarieties of ˆ A .To prove point (1), let Z be an irreducible component of V S . Note that forany e >
0, the set of P ∈ Z such that the image of H ( X, F e ∗ ( ω X ⊗ a ∗ P p e )) → H ( X, ω X ⊗ a ∗ P ) is non-zero is a constructible subset. Thus V S ⊂ V S is thecomplement of countably many locally closed subsets. (cid:3) Proposition 3.3.3.
With assumptions as in (3.3.2) (including that A has nosupersingular factors), each maximal dimensional irreducible component of theclosure of the set of points such that lim −→ H ( X, F e ∗ ω X ⊗ a ∗ P y ) ∨ = 0 is a torsiontranslate of an abelian subvariety of ˆ A .Proof. Since H ( X, ( F e ∗ ω X ) ⊗ a ∗ P ) ∼ = H ( X, F e ∗ ( ω X ⊗ a ∗ P p e )) ∼ = H ( X, F e − ∗ ( ω X ⊗ a ∗ ( P p ) p e − )) , it is easy to see that if lim −→ H ( X, ( F e ∗ ω X ) ⊗ a ∗ P ) ∨ = 0 then lim −→ H ( X, ( F e ∗ ω X ) ⊗ a ∗ P p ) ∨ = 0. The proof now follows along the lines of the previous proposition. (cid:3) Corollary 3.3.4.
With assumptions as in (3.3.2) (including that A has no super-singular factors), assume also that κ S ( X ) ≤ (See Section 4.1 for the definition)and that k is uncountable. Then V S contains at most one point.Proof. Let T + P ⊂ V S where dim T > P ∈ ˆ A is torsion and T ⊂ ˆ A is an abelian subvariety. Pick m ≥ P m ∼ = O A . Then for anyvery general Q ∈ T we have a map S ( ω X ⊗ Q ⊗ P ) m − ⊗ S ( ω X ⊗ Q − m +1 ⊗ P ) → S ( ω mX ) . It follows immediately that dim S ( ω mX ) ≥
2. This is impossible and hencedim V S = 0 and so V S is a union of finitely many torsion points. Therefore,every component of V S is zero dimensional and so V S = V S .By a similar argument to the one above it then follows that V S contains atmost one point. Suppose by way of contradiction that there are two elements P = Q in V S . By what we have seen above, P, Q ∈ ˆ A are torsion points andso there is an integer m > P m = Q m = O X . Let G P ∈ | K X + P | , G Q ∈ | K X + Q | be corresponding divisors, then mG P , mG Q ∈ | mK X | are distinctdivisors corresponding to elements of S ( ω mX ) so that dim S ( ω mX ) ≥
2. This isthe required contradiction. (cid:3)
ENERIC VANISHING AND ORDINARY ABELIAN VARIETIES 23
Proposition 3.3.5.
Let A be an abelian variety that has no supersingular factors,then each maximal dimensional irreducible component of the set Z of points P ∈ ˆ A such that the image of H (Λ ) P → Λ P is non-zero, is a torsion translate of anabelian subvariety of ˆ A and Λ P = 0 if and only if P e ∈ Z for some e > .Proof. Let V : ˆ A → ˆ A be Verschiebung so that V ([ P ]) = [ P p ] for every [ P ] ∈ ˆ A .Let V P : Spec O ˆ A,P → Spec O ˆ A,P p be the induced morphism. We have V ∗ P (Λ P p ) = V ∗ P (lim −→ ( V e, ∗ H (Λ e )) P p ) = lim −→ ( V e +1 , ∗ H (Λ e )) P = Λ P . It follows that if Λ P = 0 then also Λ P pe = 0. Let K e denote the kernel of H (Λ ) → Λ e , then K i ⊂ K i +1 ⊂ . . . so that K i = K for all i ≫
0. Therefore,the image of H (Λ ) → Λ is a coherent sheaf (isomorphic to H (Λ ) /K ). Itfollows that if Z is the set of P ∈ ˆ A such that the image of H (Λ ) P → Λ P =lim −→ ( V e, ∗ H (Λ ) P ) is non-zero, then Z is a closed subset of ˆ A . Since V is faithfullyflat, so is V P . Thus if the map H (Λ ) P → ( V ∗ H (Λ )) P = V ∗ P ( H (Λ ) P p ) → V ∗ P (Λ P p ) = Λ P is non-zero then the map H (Λ ) P p → Λ P p is non-zero as well. It follows that thatif P ∈ Z , then P p ∈ Z . If A has no supersingular factors, then the claim followsfrom (2.3.6). Finally, since Λ P = 0 if and only if the image of H (Λ e ) P → Λ P isnon-zero for some e ≥
0, it follows that Λ ⊗ O ˆ A,P = 0 if and only if P e ∈ Z forsome e ≥ (cid:3) Examples.
We begin by showing that for an ordinary abelian variety A and for an integer 0 ≤ i ≤ dim A , V i (Ω) = ˆ A where Ω = lim ←− Ω e , Ω e = F e ∗ ω A and V i (Ω) := { [ P ] ∈ ˆ A | H i ( A, Ω ⊗ P ) = 0 } . Proposition 3.4.1.
For an ordinary abelian variety A , we have { [ P ] ∈ ˆ A |∃ e > P p e ∼ = O A } = V i (Ω) Note that this is a countably infinite set by [Mumford70, Application 2, page 62] .Proof.
For every [ P ] ∈ ˆ A , H i ( A, Ω ⊗ P ) = lim ←− H i ( A, Ω e ⊗ P ) | {z } [EGA III, 13.3.1] = lim ←− H i ( A, ω A ⊗ P p e ) . In particular, if P is p -power torsion thenlim ←− H i ( A, ω A ⊗ P p e ) ∼ = lim ←− H i ( A, ω A ) | {z } discarding finitely manyterms and using that P p e ∼ = O A for every e ≫ ∼ = (lim −→ H i ( A, O A )) ∨ | {z } Serre duality where the map between the countably many copies of H i ( A, O A ) in the last directlimit is the natural homomorphism induced by the Frobenius. In particular, thishomomorphism is bijective, because A is ordinary. Hence,lim ←− H i ( A, ω A ⊗ P p e ) ∼ = H i ( A, O A ) ∨ = 0 . If P is not p e torsion for any e ≥
0, then H i ( A, ω A ⊗ P p e ) = 0 for all e ≥ H i ( A, Ω ⊗ P ) = 0. This concludes our proof. (cid:3) Proposition 3.4.2. [MvdG, (5.30)]
In the situation of the above proposition, V i (Ω) is dense in ˆ A . Hence the following seems to be the most natural question.
Question 3.4.3. Is V i (Ω) a countable union of Zariski closed sets with codimen-sion at least i ? We now compute examples of Λ.
Example 3.4.4.
Let E be an elliptic curve and Ω := ω E . There are two cases:If E is supersingular (which is equivalent for elliptic curves to being not ordi-nary), then note that R ˆ S ( O E [ g ]) = k ˆ E = Λ and R ˆ S ( F ∗ O E [ g ]) = Λ = ˆ F ∗ Λ is Artinian of length p . By (3.2.1), Λ ⊗ k ˆ E ∼ = k ˆ E . So, Λ is an Artinianˆ O ˆ E, ∼ = k [[ x ]] module which has dimension one when tensored with the residuefield. Therefore, Λ ∼ = k [[ x ]] / ( x p ) as a k [[ x ]] module. Similarly Λ e ∼ = k [[ x ]] / ( x p e ).Now, let us determine the map Λ → Λ . It is a k [[ x ]]-module homomorphism k → k [[ x ]] / ( x p ). Up to a multiplication by a unit (which can be disregarded forour purposes) there are two such maps: the zero map, and the multiplicationby x p − . Since Λ e → Λ e +1 is obtained by applying ˆ F e, ∗ to Λ → Λ , if thelatter was zero, then all the maps Λ e → Λ e +1 would be zero, and consequentlyalso Λ would be zero. This is impossible by (3.1.4), because Ω is not zero in oursituation. Hence Λ → Λ has to be the multiplication by x p − map. In particularit is injective. Since ˆ F e is faithfully flat, Λ e → Λ e +1 then has to be also injective.Therefore, Λ e → Λ e +1 can be identified with the map k [[ x ]] / ( x p e ) → k [[ x ]] / ( x p e +1 )given by multiplication by x p e ( p − . The quasi-coherent sheaf Λ is then the directlimit of the modules k [[ x ]] / ( x p e ) viewed as a direct system via multiplications by x p e ( p − . Note that this is a torsion k [[ x ]]-module, and Λ ⊗ k [[ x ]] k ˆ E = 0 (thoughΛ = 0). Further, Supp Λ = { ˆ E } .If E is an ordinary elliptic curve, then the induced map H ( O E ) → H ( F ∗ O E )is an isomorphism and ˆ F = V is the ´etale map that sends each Q ∈ ˆ E to Q p .It follows that Λ e = L Q pe = O E k ( Q ), and the maps Λ e → Λ e +1 are the naturalembeddings. Therefore Λ = L y ∈ E [ p ∞ ] k ( y ) where E [ p ∞ ] denotes the set of all p ∞ torsion points in ˆ E . In particular, Supp Λ = E [ p ∞ ], which is a countable infiniteset. ENERIC VANISHING AND ORDINARY ABELIAN VARIETIES 25 Geometric consequences
Frobenius stable Kodaira dimension.
In this section X is always asmooth, projective variety over k . In characteristic p > S ( X, O X ( mK X ))is better behaved than H ( X, O X ( mK X )). So we define S ( K X ) = M m ≥ S ( X, O X ( mK X )) ⊂ R ( K X ) = M m ≥ H ( X, O X ( mK X ))and κ S ( X ) = max { k | dim S ( X, O X ( mK X )) = O ( m k ) for m sufficiently divisible } . It is easy to see that S ( K X ) is a birational invariant. Lemma 4.1.1. S ( K X ) ⊂ R ( K X ) is an ideal.Proof. If f ∈ S ( X, O X ( mK X )) and g ∈ H ( X, O X ( lK X )) then f g ∈ S ( X, O X (( m + l ) K X )) in fact there are f e ∈ H ( X, O X ((1+( m − p e ) K X )) such that Φ e F e ∗ ( f e ) = f and by the projection formula we have f e g p e ∈ H ( X, O X ((1+( m + l − p e ) K X ))such that Φ e F e ∗ ( f e g p e ) = f g . (cid:3) Remark 4.1.2.
Note that for curves we have S ( K P ) = 0, S ( K E ) = 0 (resp. S ( K E ) = k [ x ]) if E is a supersingular (resp. ordinary) elliptic curve and if X is a curve of genus at least two, then S ( K X ) n = R ( K X ) n for n ≫
0, since K X is ample [Pat12, Corollary 2.23]. In higher dimensions, assuming the finitegeneration of R ( K X ), it then follows that the ideal S ( K X ) is a finitely generated R ( K X ) module. In particular this holds in dimension 2. Lemma 4.1.3. If κ S ( X ) ≥ , then κ ( X ) = κ S ( X ) .Proof. Since S ( X, O X ( mK X )) ⊂ H ( X, O X ( mK X )) the inequality κ ( X ) ≥ κ S ( X )is clear. The reverse inequality is immediate from the fact that S ( K X ) is a torsionfree module over the integral domain R ( K X ) and hence there are many (module)embeddings R ( K X ) ֒ → S ( K X ). (cid:3) Remark 4.1.4.
Note however that if Y is a supersingular elliptic curve, then κ S ( Y ) = −∞ but κ ( Y ) = 0. If Z is a variety of general type, then for X = Y × Z , k ( X ) = dim X −
1. However S ( X, ω mX ) = S ( Y, ω mY ) ⊗ S ( Z, ω mZ ) (c.f., [Pat12,Lemma 2.31]), and so S ( X, ω mX ) = 0 for every m >
0. Thus κ S ( X ) = −∞ and κ ( X ) = dim X − Lemma 4.1.5.
We have lim dim S ( X, O X ( mK X )) /m n = lim dim H ( X, O X ( mK X )) /m n . In particular κ ( X ) = dim X iff κ S ( X ) = dim X .Proof. If κ ( X ) = dim X , then there is a very ample line bundle A such that S ( X, A ) = 0 and an integer N >
0, such that A ֒ → ω NX . Therefore, multiplyingby a nonzero section in S ( X, A ), we have H ( X, ω mX ) ֒ → S ( X, ω mX ⊗ A ) ֒ → S ( X, ω m + NX ) . We thus have inequalities h ( X, ω mX ) ≤ dim S ( X, ω m + NX ) ≤ h ( X, ω m + NX )and the claim follows immediately. (cid:3) Lemma 4.1.6.
The limit lim || m → + ∞ dim S ( X, O X ( mK X )) m κ S ( X ) exists. (Here we have assumed that || m i.e. that m > is sufficiently divisible.)Proof. If dim S ( X, O X ( mK X )) = 0 for all m ≥ S ( X, O X ( mK X )) > m ≥
0. Arguingas in the proof of (4.1.5), we have h ( X, ω mX ) ≤ dim S ( X, ω m + NX ) ≤ h ( X, ω m + NX )for all m, N > || m → + ∞ h ( X, O X ( mK X )) m κ S ( X ) exists. (cid:3) Proof of point (1) of Theorem 1.1.1.Definition 4.2.1.
Let X be a smooth projective variety over k such that κ ( X ) =0. The Calabi-Yau index is then defined to be the greatest common divisor ofthe integers m >
0, for which h ( mK X ) = 0. By the following remark, it is alsothe smallest integer m >
0, for which h ( mK X ) = 0. Remark 4.2.2.
From the Chinese remainder theorem it follows that if r is theCalabi-Yau index of X , then there is an integer l >
0, such that h ( rlK X ) = 0and h ( r ( l + 1) K X ) = 0. Let D ∈ | rlK X | and D ′ ∈ | r ( l + 1) K X | be the uniqueelements for some l ≫
0. Then ( l + 1) D = lD ′ . In particular l +1 l D = D ′ andhence D ′ − D = l D is an effective Z -divisor, an element of | rK X | . Remark 4.2.3.
Note also that if κ S ( X ) = 0, then κ ( X ) = 0 by Lemma 4.1.3,and hence the Calabi-Yau index is defined. Lemma 4.2.4. If X is a smooth projective variety with κ S ( X ) = 0 , then theCalabi-Yau index of X divides p − .Proof. Let and m > S ( X, ω mX ) = 0 and r be theCalabi-Yau index, then r | m . Furthermore, since for every e > H ( X, O X (( p e m + (1 − p e )) K X )), where0 ≡ p e m + (1 − p e ) ≡ (1 − p e ) mod r. For e = 1 we obtain that r | p − (cid:3) ENERIC VANISHING AND ORDINARY ABELIAN VARIETIES 27
Lemma 4.2.5. If X is a smooth projective variety such that κ S ( X ) = 0 withCalabi-Yau index r and a : X → A is morphism to a projective scheme A over k , then (1) S ( X, ω rX ) = 0 , (2) if G ∈ | rK X | is the unique element, then S ( X, σ ( X, ∆) ⊗ ω rX ) = 0 , where ∆ = r − r G , (3) the natural inclusions S ( X, σ ( X, ∆) ⊗ ω rX ) ⊆ H ( A, Ω ) ⊆ H ( X, ω rX ) are equalities, where Ω = S a ∗ ( σ ( X, ∆) ⊗ ω rX ) and ∆ is as above and thenatural inclusion is obtained from (2.2.2) . (4) the natural maps H ( A, F e +1 ∗ Ω ) → H ( A, F e ∗ Ω ) and H ( X, F e +1 ∗ ω rX ) → H ( X, F e ∗ ω rX ) are compatible with the above inclusions and hence both areisomorphisms.Proof. For point (1), let 0 = f ∈ H ( X, O X ( rK X )) corresponding to a divisor G . Then, there is an integer l >
0, such that S ( X, ω rlX ) = 0 (4.1.3). Thus,for all integers e >
0, there is an element of H ( X, O X (( p e rl + (1 − p e )) K X ))mapping to 0 = f l ∈ H ( X, O X ( rlK X )). Since κ ( X ) = 0, that element can onlybe αf p e l + − per for some α ∈ k ∗ . Now, let us look at Φ e (cid:16) F e ∗ (cid:16) αf p e + − per (cid:17)(cid:17) . If itwere zero, then the following element would also be zeroΦ e (cid:16) F e ∗ (cid:16) αf p e + − per (cid:17)(cid:17) f l − = Φ e (cid:16) F e ∗ (cid:16) αf lp e + − per (cid:17)(cid:17) = f l . But we know that f l is not-zero, so also Φ e (cid:16) F e ∗ (cid:16) αf p e + − per (cid:17)(cid:17) is not zero andhence equals βf for some β ∈ k ∗ . Therefore, f ∈ im Φ e for every e > f ∈ S ( X, O X ( rK X )).Point (2) follows from the fact that that the image of αf ∈ H ( X, O X ( rK X )) = H ( X, O X ( rp e K X + (1 − p e )( K X + ∆)))in H ( X, O X ( rK X )) is computed by αf αf · f r − r ( p e − = αf p e + − per βf. Point (3) immediately follows from the fact that S ( X, ω rX ) = 0, and thatdim k H ( X, ω rX ) = 1.Point (4) follows from H ( A, ) applied to the commutative diagram F e +1 ∗ Ω / / (cid:127) _ (cid:15) (cid:15) F e ∗ Ω (cid:127) _ (cid:15) (cid:15) F e +1 ∗ a ∗ ω rX / / F e ∗ a ∗ ω rX a ∗ F e +1 ∗ ω rX / / a ∗ F e ∗ ω rX , where the bottom two horizontal arrows are the arrows from (2.2.3). Further thebottom one is an isomorphism because the stable image of these maps is exactly S ( X, σ ( X, ∆) ⊗ ω rX ), which is proven to be non-zero in point (2). (cid:3) Theorem 4.2.6.
Let a : X → A be a generically smooth morphism from asmooth projective variety to (but not necessarily onto) an abelian variety withgeneral fiber G . If S ( G, ω G ) = 0 , then H ( X, ω X ⊗ a ∗ P ) > for some P ∈ ˆ A .Proof. Since S ( G, ω G ) = 0, it follows by (2.5.1) that S a ∗ ω X = 0. If H ( X, ω X ⊗ a ∗ P ) = 0 , ∀ P ∈ ˆ A, then Ω = 0 by (3.2.2), but since Ω → S a ∗ ω X = 0 is surjective, this is impossible. (cid:3) Remark 4.2.7.
Note that the above theorem does not hold if one replaces H by S . That is, there are examples where S ( X, ω X ⊗ a ∗ P ) = 0 for every P ∈ ˆ A .An easy example is when X = A is a non-ordinary abelian variety, and a = id A .Then S ( X, ω X ⊗ a ∗ P ) = S ( A, ω A ⊗ P ), which is zero for P = O A because then H ( A, ω A ⊗ P ) = 0 and it is zero for P = O A , because A is not ordinary. Theorem 4.2.8.
Let X be a smooth projective variety defined over k and a : X → A the Albanese morphism. Suppose that S a ∗ ω X = 0 and Pic ( X ) has nosupersingular factors, then if κ ν ( X ) = 0 , then κ ( X ) = 0 .Proof. By [CHMS12], we know that K X ≡ G for the effective R -divisor G = N σ ( K X ). Assume now that P ′ ∈ V ( ω X ), that is, h ( K X + a ∗ P ′ ) = 0. ThenFix( K X + a ∗ P ′ ) ≥ ⌈ N σ ( K X + a ∗ P ′ ) ⌉ = ⌈ N σ ( K X ) ⌉ = ⌈ G ⌉ , and hence h ( K X + a ∗ P ′ − ⌈ G ⌉ ) = 0. In particular, K X + a ∗ P ′ − ⌈ G ⌉ ∼ E for some effective divisor E . Therefore, using that G ≡ K X , for the effective R -divisors E and {− G } wehave E + {− G } ≡
0. This implies that both E = 0 and {− G } = 0. In particular, G is an integral divisor. Hence, we may assume that K X − G ∼ Q + a ∗ P forsome torsion divisor Q and P ∈ ˆ A . We have that K X + a ∗ P ′ − G ∼ E as earlier,where E = 0 by the same argument as before. Hence 0 ∼ Q + a ∗ ( P + P ′ ). Thisimplies that Q ∈ Pic ( X ). So, we may assume that Q = 0 and then P = − P ′ .We obtained that V ( ω X ) = {− y } where P = P y . Further, we have V ( S a ∗ ω X ) = V ( ω X ) ⊂ {− y } . It follows that the support of H (Λ ) is contained in y whereΛ = R ˆ S ( D A ( S a ∗ ω X )). By (3.3.5), the image of H (Λ ) → Λ is a finite unionof torsion translates of subtori of ˆ A . Note that this image is not 0 as otherwiseΛ = 0 and hence Ω = 0 (which is impossible as Ω surjects on to S a ∗ ω X ). Thus y is a torsion point. (cid:3) Definition 4.2.9. If X is smooth, projective over k , then the i -th Betti number b i ( X ) is defined as dim Q l H i ´et ( X, Q l ) for some l = p . Remark 4.2.10.
Note that the above definition of b i ( X ) is independent of thechoice of l by an application of [Deligne74]. Furthermore, b ( X ) = 2 dim Alb( X )by [Liedtke09, page 14]. ENERIC VANISHING AND ORDINARY ABELIAN VARIETIES 29
Theorem 4.2.11.
Let a : X → A be the Albanese morphism of a smooth projec-tive variety over k . If κ S ( X ) = 0 , then a : X → A is surjective. In particular, if κ S ( X ) = 0 , then b ( X ) ≤ X .Proof. Let r be the Calabi-Yau index of X . By Lemma 4.2.5 there is a unique G ∈ | rK X | . We use the notations of (3.1.3) with the definitions∆ := r − r G, Ω e := F e ∗ S a ∗ ( σ ( X, ∆) ⊗ O X ( rK X )) and Ω := lim ←− Ω e . Note that by (4.2.4), r | ( p e −
1) for every e ≥ s of(2.2.3) (or of (3.1.3)) to be 1. Also, according to (2.2.3), Ω e +1 → Ω e is surjectivefor every integer e ≥
0. Further, by (4.2.5), S a ∗ ( σ ( X, ∆) ⊗ O X ( rK X )) = 0.Therefore, for every e , Ω e = 0 and hence Ω = 0.We claim that since κ ( X ) = 0, there is a neighborhood U of the origin suchthat V ( ω rX ) ∩ U = { P ∈ ˆ A | h ( X, ω rX ⊗ a ∗ P ) = 0 } ∩ U = { ˆ A } . Suppose that this is not the case and let T ⊂ ˆ A be a positive dimensionalirreducible component of V ( ω rX ) through the origin. Let ξ : T g +1 → A be thenatural morphism. By dimension count, every fiber of ξ is positive dimensional.Further, since 0 ∈ T , 0 is in the image of ξ . Consider then for ( P , . . . , P b +1 ) ∈ ξ − ( O A ) the maps H ( X, ω rX ⊗ a ∗ P ) ⊗ . . . ⊗ H ( X, ω rX ⊗ a ∗ P b +1 ) → H (cid:16) X, ω ( b +1) rX (cid:17) . Since dim H (cid:16) X, ω ( b +1) rX (cid:17) = 1, there are only finitely many choices for the di-visors in | rK X + a ∗ P i | for P i ∈ p ( ξ − ( O A )). However, since p ( ξ − ( O A )) is aninfinite set of not isomorphic line bundles and Pic A → Pic X is injective, this isa contradiction. This finishes the proof of our claim.Since H ( A, S a ∗ ( σ ( X, ∆) ⊗ ω rX ) ⊗ P ) ⊂ H ( X, ω rX ⊗ a ∗ P ), we obtain that H ( A, Ω ⊗ P ) is zero for every O A = P ∈ U . Further, H ( A, Ω ) = 0 by(4.2.5). Hence, by (3.2.1), H (Λ ) is Artinian in a neighborhood of the originand H (Λ ) ⊗ k (0) = 0. Therefore, H (Λ ) ∼ = C ⊕ B , where B is Artinian andsupported at 0 and 0 Supp C . This induces a similar decomposition C e ⊕ B e on H (Λ e ) ∼ = V ∗ H (Λ ). Furthermore, the map H (Λ e ) → H (Λ e +1 ) is the directsum of morphisms C e → C e +1 and B e → B e +1 . HenceΛ = lim −→ Λ e = (cid:0) lim −→ C e (cid:1) ⊕ (cid:0) lim −→ B e (cid:1) , where B e is Artinian for every e ≥ We claim that lim −→ B e = 0 . We argue this by showing that (lim −→ B e ) ⊗ k (0) =lim −→ ( B e ⊗ k (0)) is not zero. To this end note that B e ⊗ k (0) ∼ = H ( A, F e ∗ Ω ) ∨ ,and the homomorphism B e ⊗ k (0) → B e +1 ⊗ k (0) can be identified via theseisomorphisms with H ( A, F e ∗ Ω ) ∨ → H ( A, F e +1 ∗ Ω ) ∨ . However, the latter is anisomorphism by (4.2.5). Hence lim −→ ( B e ⊗ k (0)) = 0 and consequently lim −→ B e = 0,which concludes the proof of our claim. Finally, (3.2.3), concludes our proof. (cid:3)
Proof of point (2) of Theorem 1.1.1.Theorem 4.3.1.
Let X be a smooth projective variety over k and a : X → A the Albanese morphism. Then the following are equivalent: (1) p ∤ deg a , κ S ( X ) = 0 and b ( X ) = 2 dim X , and (2) X is birational to an ordinary abelian variety.Proof. It suffices to show that (1) implies (2). Assume that κ S ( X ) = 0 anddim X = dim A . Then a is surjective by (4.2.11) and hence it is genericallyfinite. We must show that the generic degree of a is 1 and A is ordinary. We setΩ = S a ∗ ω X , Ω e = F e ∗ Ω , Ω = lim ←− Ω e , Λ e = R ˆ S ( D A (Ω e )) and Λ = hocolim −→ Λ e asin (3.1.3). Step 1. S ( ω X ) = 0 , A is ordinary and the inclusion ω A → a ∗ ω X factorsthrough the embedding S a ∗ ω X → a ∗ ω X .In fact, in this step, we do not need to assume that p ∤ deg a , only that a isseparable, hence the ordinarity of A holds in this more general context. Since a is separable, there is a natural inclusion ω A → a ∗ ω X (the dual of the trace map a ∗ O X → O A ) inducing an inclusion H ( ω A ) ⊂ H ( ω X ). Since κ ( X ) = κ S ( X ) =0, this inclusion is an equality, and by (4.2.5) S ( ω X ) = 0. We claim that theinclusion ω A → a ∗ ω X is compatible with Frobenius in the sense that the followingdiagram commutes . F ∗ ω A (cid:15) (cid:15) / / ω A (cid:15) (cid:15) F ∗ a ∗ ω X / / a ∗ ω X Dualize the above diagram, that is apply H om O A ( , ω A ) to it (and Grothendieckduality at multiple places). Since dualization applied twice is the identity, it isenough to show that the dualized diagram commutes: F ∗ O A O A o o F ∗ a ∗ O X O O a ∗ O X o o O O . Because F ∗ O A is reflexive, it is further enough to show that the above diagramcommutes in codimension one. That is, we may assume that a is a finite mapof normal varieties. Let φ i be the embeddings of the function field of X intoits algebraic closure over the function field of A . Then we have to verify that( P φ i ( f )) p = P φ i ( f p ) holds for every local section f of O X . However, this fol-lows since φ i are ring homomorphisms and hence φ i ( f p ) = φ i ( f ) p . This concludesour claim. ENERIC VANISHING AND ORDINARY ABELIAN VARIETIES 31
Since the inclusion ω A → a ∗ ω X is compatible with Frobenius, it follows that S ( ω A ) ∼ = S ( ω X ) = 0. By (2.3.2), this is equivalent to saying that A is ordinary.Further, since S ( X, ω X ) ⊆ H ( A, S a ∗ ω X ) ⊆ H ( A, a ∗ ω X ) ∼ = H ( X, ω X ) , by (2.2.2) and since ω A ∼ = O A , we also see that the natural morphism ω A → a ∗ ω X factors through Ω = S a ∗ ω X . Step 2. V ( ω X ) contains no torsion points except O A . Assume the contrary, i.e., let Q = O A be a torsion point of V ( ω X ) and let D be the unique element of | K X | . Then considering the multiplication map (where o ( Q ) is the order of Q ), | K X + Q | × · · · × | K X + Q | → | o ( Q ) K X | we see that the only element in | K X + Q | can be o ( Q ) o ( Q ) D = D . Hence K X + Q ∼ D ∼ K X , which is a contradiction. Step 3.
The image of Λ → Λ is supported on ˆ A . If this were not the case, then by (3.3.5), there is a torsion translate 0 ˆ A = T ofan abelian subvariety such that for each Q ∈ T , the map Λ ⊗ O ˆ A,Q → Λ ⊗ O ˆ A,Q is non-zero. Then by (3.2.1) also T ⊆ V (Ω ) ⊆ V ( ω X ), and by Step 2 oneobtains a contradiction. Step 4. H (Λ ) ∼ = k (0) . Since we assume that p ∤ deg a , the embedding ω A ֒ → a ∗ ω X is in fact a splitting.Further since this map factors through ω A ֒ → Ω , the latter also splits. Inparticular, H (Λ ) has a direct summand isomorphic to H ( R ˆ S ( D A ( ω A ))) ∼ = k (0).However, by (3.2.1) dim k H (Λ ) ⊗ k (0) = 1. So, any direct complement F of k (0)in H (Λ ) is a coherent sheaf supported at 0, such that F ⊗ k (0) = 0. Therefore, F = 0. Step 5. deg a = 1 . By (3.1.1), we know that there is a factorizationΛ / / ) ) H (Λ ) / / Λ . By Steps 4, H (Λ ) ∼ = k (0). Thus we have a commutative diagramΛ / / ( ( k (0) / / Λ . Applying ( − A ) ∗ D A ( RS ( ))[ − g ] to the above diagram we obtainΩ O A o o Ω v v o o . Since the long arrow is surjective we obtain that rk Ω = 1. This concludes theproof. (cid:3) References [BB11] M. Blickle, and G. B¨ockle,
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E-mail address : [email protected] Department of Mathematics, Princeton University, Fine Hall, WashingtonRoad, NJ 08544-1000, USA
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