aa r X i v : . [ m a t h . AG ] M a r ON VARIETIES WITH TRIVIAL TANGENT BUNDLE
KIRTI JOSHI
Abstract.
I prove a crystalline characterization of abelian varieties in characteristic p > p >
And here I stand, with all my lore,Poor fool, no wiser than before.Goethe, Faust part I Introduction
Let k be a field and let X be a smooth projective variety over k . For k = C it is well-known,and elementary to prove, that if X has trivial tangent bundle, then X is an abelian variety.In [Igusa, 1955] it was shown that this is false in characteristic p >
0. [Mehta and Srinivas,1987] studied ordinary varieties with trivial tangent bundle and proved that they have manyproperties similar to abelian varieties, including the Serre-Tate theory of canonical liftings.In Theorem 2.4, I present two equivalent crystalline characterizations of abelian varietiesamongst the class of varieties with trivial tangent bundle. My characterization is the fol-lowing: a smooth, projective variety X with trivial tangent bundle is an abelian variety ifand only if it has a smooth Picard scheme and it satisfies Hodge symmetry in dimensionone (I call such a variety Picard-Hodge Symmetric , see Def 2.3). Another equivalent char-acterization is given in terms of what I call minimally Mazur-Ogus varieties (see Def 2.1).A smooth, projective variety is a minimally Mazur-Ogus variety if H cris ( X/W ) is torsion-free and Hodge de Rham spectral sequence degenerates in dimension one. In Corollary 2.11I show that any smooth, projective variety with trivial tangent bundle which lifts to W and with H cris ( X/W ) torsionfree is an abelian variety. In Remark 2.13 I discuss a naturalquestion raised by Li in his emails to me about weakening the hypothesis of Theorem 2.4.In [Li, 2010, Conjecture 4.1] it is conjectured that if p > p = 2 with trivialtangent bundle and which is not an abelian variety is due to [Igusa, 1955] (Igusa surface for p = 2 has been studied by many authors including Torsten Ekedahl; for a recent treatmentof the Igusa surface see [Chai]). Let me note that the Igusa surface of characteristic p = 2also has a less well-known cousin in characteristic p = 3. I observe in Theorem 3.6 that if p = 2 then for every g ≥ ≤ r < g , thereis a family of varieties dimension g with trivial tangent bundle and which are not abelianvarieties. This family is parameterized by A ord r [ p ] × A g − r where A g is the moduli stack ofabelian varieties of dimension g and the superscript ‘ord’ stands for the “ordinary locus” and A ord r [ p ] is the moduli stack of ordinary abelian varieties with a point of order p . For p = 3one has a slightly weaker result–see Theorem 3.8.In Remark 3.9, I note that the two conditions: minimally Mazur-Ogus, Picard-Hodgesymmetry in Theorem 2.4 cannot be weakened or relaxed. In general, presence of torsion incrystalline cohomology and non-degeneration of Hodge de Rham are not correlated condi-tions.In Theorem 4.1, I show that a smooth, projective, ordinary variety with trivial tangentbundle is an abelian variety if and only if its second crystalline cohomology is torsion free.In [Li, 2010, Theorem 4.2] (also see [Li, 1991]) it is shown that if p > X is ordinarywith trivial tangent bundle then X is an abelian variety. In Theorem 5.2, I give a new proofof Li’s remarkable theorem [Li, 2010, Theorem 4.2] and in fact I prove a sharpening of [Li,2010, Theorem 4.2] and of [Mehta and Srinivas, 1987]. I show that for p = 2 any smooth,projective, ordinary variety with trivial tangent bundle has a minimal Galois ´etale cover (seeDef. 5.1) by an abelian variety with Galois group of exponent p = 2. Li’s approach is basedon infinitesimal group actions while I use Serre-Tate canonical liftings (of abelian varieties)and the theory of complex multiplication and its influence on the slopes of Frobenius (see[Yu, 2003]).In the light of my characterization (Theorem 2.4), especially as torsion in the secondcrystalline cohomology can occur for any prime p , it seems to me that perhaps the originalconjecture of Li (see [Li, 2010, Conjecture 4.1]) needs to be modified. In fact there aretwo different versions of Li’s conjecture which I conjecture. The first version is the fixedcharacteristic version which says that there exists an integer n ( p ) such that if X is anyvariety of dimension less than n ( p ), with trivial tangent bundle over an algebraically closedfield of characteristic p > d ≥
2. There exists an integer n ( d ) such that any smooth, projective variety X/k with dimension d and with trivial tangent bundle is an abelian variety if p > n ( d ).(Clearly for d = 1, one has n (1) = 1; for d = 2, n (2) = 3 by Theorem 3.1).I would like to thank Vikram Mehta for bringing [Li, 2010] to my attention and for manyconversations around Li’s conjecture. Many years ago (around 1991-92) Vikram had ex-plained to me his paper with V. Srinivas (see [Mehta and Srinivas, 1987]), and more recentlyduring my visit to India in 2012 we had many lucid conversations on topics of commoninterest (we were studying some questions on fundamental group schemes). I was amazedby his incredible energy and zest for mathematics while facing an illness which ultimatelytook him from us.I proved Theorem 2.4 while I was on a visit to RIMS, Kyoto in 2011. I thank RIMS, Kyotofor providing excellent hospitality and I am grateful to Shinichi Mochizuki providing me withan opportunity to visit RIMS. I thank KeZheng Li for answering many of my elementaryand naive questions about his papers [Li, 1991, 2010], and for his comments and corrections.I thank Brian Conrad for pointing out [Yu, 2003]. Characterization of abelian varieties
In this section I give a crystalline characterization of abelian varieties in the class ofsmooth, projective varieties with trivial tangent bundle. My characterization requires thefollowing two definitions.
Definition 2.1.
Let X be a smooth, projective variety over an algebraically closed field k with char( k ) = p >
0. I say that X is a minimally Mazur-Ogus variety if X satisfies thefollowing two conditions:(1) H cris ( X/W ) is torsion free,(2) Hodge to de Rham sequence degenerates in dimension one.
Remark 2.2.
Conditions underlying Mazur-Ogus varieties were introduced in [Ogus, 1979]where a number of their properties are studied, the nomenclature, I believe, is due to TorstenEkedahl. A smooth, projective variety X is a Mazur-Ogus variety if H ∗ cris ( X/W ) is torsion-free and Hodge de Rham spectral sequence degenerates.
Definition 2.3.
Let X be a smooth, projective variety over an algebraically closed field k with char( k ) = p >
0. I say that X is a Picard-Hodge symmetric variety if it satisfies thefollowing two conditions:(1) Picard scheme of X is smooth,(2) Hodge symmetry holds in dimension one.The main theorem of this section is the following characterization theorem alluded to inthe Introduction. Theorem 2.4.
Let X be any smooth, projective variety with trivial tangent bundle. Thenthe following are equivalent.(1) X is minimally Mazur-Ogus,(2) X is Picard-Hodge symmetric,(3) X is an abelian variety.Proof. Let us prove (1) ⇒ (2) ⇒ (3) ⇒ (1). Let us begin with (1) ⇒ (2). Assume that X isminimally Mazur-Ogus. The fact that H cris ( X/W ) is torsion-free implies Pic ( X ) is reduced(see [Illusie, 1979]) and by the universal coefficient theorem for crystalline cohomology onesees that(2.5) H cris ( X/W ) ⊗ W k ∼ / / H dR ( X/k ) . As Hodge to de Rham degenerates in dimension one, one sees that(2.6) dim( H dR ( X/k )) = h , + h , . Reducedness of Picard variety means that(2.7) dim( H cris ( X/W ) ⊗ W k ) = 2 h , and the degeneration of Hodge de Rham mean that(2.8) 2 h , = h , + h , . Thus one sees that(2.9) h , = h , . KIRTI JOSHI
Putting all this together one sees that X is Picard-Hodge symmetric variety. Thus one seesthat (1) ⇒ (2).Now I prove (2) ⇒ (3). Suppose that X is Picard-Hodge symmetric variety and X hastrivial tangent bundle so H ( X, Ω X ) has dimension n = dim( X ). As X is Picard-Hodgesymmetric one sees that(2.10) h , = h , = dim( X ) . Thus dim(Pic ( X )) = dim( X ) and by hypothesis of (2) Pic ( X ) is reduced. Hence the Picardvariety is also the Albanese variety: Pic ( X ) = Alb( X ) and in particulardim( X ) = dim(Pic ( X )) = dim(Alb( X )) . Let X → Alb( X ) be the Albanese morphism. By [Mehta and Srinivas, 1987, Lemma 1.4]one sees that the Albanese morphism X → Alb( X ) is a smooth surjective morphism withconnected fibres and Ω X/ Alb( X ) = 0. So X → Alb( X ) is a finite, surjective ´etale morphismwith connected fibres and hence it is an isomorphism.Now it remains to prove that (3) ⇒ (1). This is standard (see [Illusie, 1979]). (cid:3) The following corollary of [Deligne and Illusie, 1987] and Theorem 2.4 is immediate as onehas Hodge de Rham degeneration in dimensions ≤ p − p (and hence in dimensionone for any p ≥ Corollary 2.11.
Let
X/k be a smooth, projective variety with trivial tangent bundle. Suppose X satisfies the following:(1) H cris ( X/W ) is torsionfree,(2) X lifts to W .Then X is an abelian variety. Remark 2.12.
Let me point out that for the Igusa surface ( p = 2 , H cris ( X/W ) is nottorsion-free (but Hodge-de Rham degenerates in dimension one) and Hodge symmetry istrue in dimension one, but Pic ( X ) is not reduced. Remark 2.13.
In his recent email to me, KeZheng Li has suggested that perhaps, anysmooth, projective variety with trivial tangent bundle and reduced Picard scheme is anabelian variety. This is certainly natural expectation. I include some comments on thisquestion.Firstly let me point out that there are two important numbers dim( X ) = dim H ( X, Ω X )and dim(Pic ( X )) = dim H ( X, O X ) which must be equal if this assertion holds. On theother hand even if Pic ( X ) is reduced, it seems difficult to prove that these two numbersare equal without some additional crystalline torsion-freeness hypothesis. Note that thepull-back of one-forms on Pic ( X ) = Alb( X ), by X → Alb( X ), lands inside the subspace ofclosed one-forms H ( X, Z Ω X ) and all of the following inclusions H (Alb( X ) , Ω X ) ) ⊂ H ( X, Z Ω X ) ⊂ H ( X, Ω X )are strict in general. By [Illusie, 1979, Prop. 5.16, page 632] the hypothesis that H cris ( X/W )is torsionfree is equivalent to reducedness of Pic ( X ) and the equality H (Alb( X ) , Ω X ) ) = H ( X, Z Ω X ). In particular the second inclusion does not become equality even if we assume H cris ( X/W ) is torsionfree, and so it is not possible to work with a simpler hypothesis: Pic ( X )is reduced at the moment. Secondly let me point out that the reducedness of Picard scheme controls only a part of thecrystalline torsion which may arise in this situation. Torsion arising from non-reducednessof Pic ( X ) is of a fairly mild sort (“divisorial torsion” in the terminology of [Illusie, 1979]).But Ekedahl has shown that the self-product of the Igusa type surface with itself carriesexotic torsion in H . It is possible that a similar example (of dimension bigger than two)exists in which H cris ( X/W ) has exotic torsion, since there is a plethora of examples (seeTheorem 3.6) in any dimension for p = 2 and one can probably use deformation theory toprovide examples with subtler torsion behaviour.So relaxing the conditions in Theorem 2.4 seems a bit too optimistic (to me) and at anyrate requires a fuller understanding of the crystalline cohomology of varieties with trivialtangent bundles (which I do not possess).It is possible to provide alternate formulations of Theorem 2.4, but I have chosen formu-lations which are easiest to deal with in practice.3. Surfaces with trivial tangent bundle
Let
X/k be a smooth projective variety over an algebraically closed field of characteristic p >
0. The main theorem of this section is the following. This was conjectured by KeZhengLi in [Li, 2010, Conjecture 4.1].
Theorem 3.1.
Let
X/k be a smooth projective surface over an algebraically closed field ofcharacteristic p > and assume that the tangent bundle T X of X is trivial. Then X is anabelian surface.Proof. As T X = O X ⊕ O X one sees that Ω X = O X ⊕ O X and so Ω X = O X . Thus c ( X ) = 0and also as T X is trivial one sees that c ( X ) = 0. Now it is immediate by the adjunctionformula (see [Hartshorne, 1977]) that X is a minimal surface of Kodaira dimension κ ( X ) = 0.By Noether’s formula 12 χ ( O X ) = c + c (see [Hartshorne, 1977]), one sees that(3.2) χ ( O X ) = 0 . This means(3.3) χ ( O X ) = 0 = h − h , + h , ;and as K X = O X by Serre duality one sees that H ( O X ) = H ( O X ), and hence that(3.4) h , = 2 . Next c = 0 gives(3.5) c = b − b + b − b + b = 2 − b + b = 0 . Thus one sees that b = 0 and one has b = 0 because X is projective (the Chern class ofany ample class is non-zero in H et ( X, Q ℓ )). Now b is even as b is the Tate module of theAlbanese variety of X (which is reduced by definition). Thus one has b ≥ b , b ): either ( b , b ) = (4 ,
6) or ( b , b ) = (2 , X is an abelian surface.If not, one is in the second case. In this case one has b = 2 so q = 1 and h ( O X ) = 2.Thus one sees that Pic ( X ) is non-reduced and at any rate the surface X is hyperelliptic andas p >
3, classification (see [Bombieri and Mumford, 1977, Page 37]) shows that the order of
KIRTI JOSHI K X must be one of 2 , , , >
1. On the other hand one has K X = O X .Thus X cannot be hyperelliptic.So one sees that the second case cannot occur and X is an abelian surface as asserted. (cid:3) By a family of varieties with trivial tangent bundle
I mean a proper, flat 1-morphism ofstacks f : X → M , with M a Deligne-Mumford stack (over schemes over k ) such that f is schematic and for every morphism of stacks Spec( k ′ ) → M with k ′ ⊃ k a field, the fibreproduct X × M Spec( k ′ ) is a geometrically connected, smooth, projective scheme over k ′ withtrivial tangent bundle.The construction of Igusa surface ([Igusa, 1955]) leads to the following (for another variantof this construction see Proposition 5.3). For g ≥
1, let A g be moduli stack of abelian varietiesof dimension g over k (see [Faltings and Ching-Li Chai, 1980, Fogarty and Mumford]). Let A ord g be the dense open substack of ordinary abelian varieties in the moduli stack of abelianvarieties of dimension g over k , more generally let U ≥ g [ p ] ⊂ A g be the stack of abelianvarieties of dimension g , with a point of order p . Theorem 3.6.
Let k be an algebraically closed field of characteristic p = 2 . Then for every g ≥ , and for any ≤ r < g , there exists a family, parameterized by U ≥ r [ p ] × A g − r ofsmooth, projective varieties of dimension g over k which are not abelian varieties and withtrivial tangent bundles. In particular there is a family parameterized by A ord r [ p ] × A g − r , ofsmooth, projective varieties of dimension g over k which are not abelian varieties and withtrivial tangent bundles.Proof. First let me recall the following version of Igusa’s construction (see [Igusa, 1955]). Foradditional variants of Igusa’s construction see Proposition 5.3 below. Let B be an abelianvariety of dimension r over k with 2-rank at least one, let t ∈ B [2]( k ) be a non-trivial twotorsion point. Let B be any abelian variety over k of dimension g − r . Then consider theIgusa action on A = B × B → B × B given by ( x, y ) ( x + t, − y ). Then this gives anaction of Z / A which is fixed point free and(3.7) H ( A, Ω A ) Z / = H ( A, Ω A ) , as Z / B andon the second factor the action on the space of one forms of B is by − X be the quotient of A by this Z / T X is trivial (as H ( X, T X ) = H ( A, T A )). On the other hand byIgusa, Alb( X ) = B / h t i and so X is not an abelian variety and Pic ( X ) is not reduced.Now one simply has to note that one can carry out Igusa’s construction on the universalabelian scheme over the moduli stack of abelian schemes (of the above sort). (cid:3) For p = 3 the result is a little weaker, by simply taking products with an abelian varietyone gets: Theorem 3.8.
Let p = 3 and k be an algebraically closed field of characteristic p . Then forevery g ≥ , there exists a family of smooth, projective varieties of dimension g over k whichare not abelian varieties and with trivial tangent bundle. Remark 3.9.
Note that for the Igusa surface one has dim H ( X, Ω X ) = dim H ( X, O X ) soHodge symmetry holds and Hodge-de Rham does degenerate at E but Pic ( X ) is not reducedand H cris ( X/W ) has torsion. Varieties X , constructed as in Theorem 3.6 from ordinaryabelian varieties, have the property that they are ordinary with trivial tangent bundle; one has lifting to W (by [Joshi, 2007, Theorem 9.1] of V. B. Mehta) and hence Hodge-deRham degenerates in dimension < p (by [Deligne and Illusie, 1987]), but H cris ( X/W ) is nottorsion-free. Thus these varieties are neither Picard-Hodge symmetric nor are they minimallyMazur-Ogus. 4.
Ordinary varieties with trivial tangent bundle
I give a proof of the following theorem.
Theorem 4.1.
Let X be a smooth, projective variety with trivial tangent bundle. Then thefollowing are equivalent: (1) X is ordinary and minimally Mazur-Ogus, (1 ′ ) X is ordinary and Picard-Hodge symmetric, (2) X is Frobenius split and minimally Mazur-Ogus, (2 ′ ) X is Frobenius split and Picard-Hodge symmetric, (3) X is ordinary and H cris ( X/W ) is torsion-free, (3 ′ ) X is Frobenius split and H cris ( X/W ) is torsion-free, (4) X is an ordinary abelian variety.Proof. The equivalences (1) ⇐⇒ (1 ′ ) and (2) ⇐⇒ (2 ′ ) are clear from the proof ofTheorem 2.4. The equivalence (3) ⇐⇒ (3 ′ ) is [Mehta and Srinivas, 1987, Lemma 1.1]. Theequivalence (1) ⇐⇒ (2) is immediate from [Mehta and Srinivas, 1987, Lemma 1.1] as X isordinary if and only if X is Frobenius split. Now (2) = ⇒ (3) is clear from the Definition 2.1and by [Mehta and Srinivas, 1987]. Now to prove (3) = ⇒ (4). This is immediate fromTheorem 2.4, provided one proves that Hodge de Rham spectral sequence degenerates indimension ≤
1. In other words one has to show that the hypothesis of (3) implies that X isminimally Mazur-Ogus. This is proved as follows. Any smooth, projective variety with trivialtangent bundle is ordinary if and only if it is Frobenius split (see [Mehta and Srinivas, 1987,Lemma 1.1]). A result of V. B. Mehta (see [Joshi, 2007, Theorem 9.1]) says that a Frobeniussplit variety X lifts to W and hence Hodge de Rham degenerates in dimension ≤ p − p ≥
2. Hence hypothesis of (3) imply that X is Mazur-Ogus. So the assertion (3) = ⇒ (4) follows from Theorem 2.4. Now (4) = ⇒ (1) is standard (see [Illusie, 1979]). (cid:3) Corollary 4.2.
Let X be a smooth, projective, ordinary variety, with trivial tangent bundle.Then X is an (ordinary) abelian variety if and only if H cris ( X/W ) is torsion-free. New proof of Li’s Theorem
In this section I give a new proof of Li’s Theorem (see [Li, 1991, 2010]) and prove thefollowing refinement.
Definition 5.1.
Let X be a smooth, projective variety with trivial tangent bundle andsuppose A → X is Galois an ´etale cover by an abelian variety. I say that A → X is aminimal Galois ´etale cover of X is there is no factorization of A → X into ´etale morphisms A → A ′ → X with A ′ an abelian variety and A ′ → X Galois.Since an abelian variety A has a non-trivial fundamental group, non-minimal ´etale coversexist if G = { } . KIRTI JOSHI
Theorem 5.2.
Let k be an algebraically closed field of characteristic p > . Let X/k be asmooth, projective, ordinary variety with trivial tangent bundle.(1) Either X is an abelian variety, or(2) p = 2 and X has a minimal Galois ´etale cover by an abelian variety with Galois groupof exponent p (i.e every element is of order p ).Proof. Let X be as in the statement of the theorem and suppose X is not an abelian variety.By [Mehta and Srinivas, 1987] there exists an ordinary abelian variety A/k and a finite,Galois ´etale morphism A → X with Galois group G of order a power of p which acts freelyon X . By passing to a quotient of G if needed, one may assume that A → X is a minimalGalois ´etale cover of X . In particular A carries fixed point-free automorphisms σ : A → A of order d = p m a power of p . If d = 1 for every element of G then this is already case (1) sothere is nothing to do; if d = 2 for every element of G then one is in case (2) so again thereis nothing to prove. So assume d = p m ≥ σ ∈ G .Then, by [Lange, 2001, Lemma 3.3] (the proof given there is characteristic free–and theargument is sketched below for convenience), there are abelian varieties A , A such that A is isogenous to A × A and that σ | A is a translation, and σ | A is an automorphism (possiblywith fixed points) of order a power of d . Indeed, write σ = t x ◦ σ ′ where t x is a translation, σ ′ an automorphism of order a power of d and one may take A to be the connected componentof ker(1 − σ ′ ) and A = image(1 − σ ′ ). As A is ordinary, so are A and A . One assumes,without loss of generality, that σ ′ is a homomorphism of A . Now ( A , σ ′ ) admits a canonicalSerre-Tate lifting to W ( k ) (see [Mehta and Srinivas, 1987, Theorem 1(2) of Appendix]), andin particular a lifting ( B , σ ′ ) of ( A , σ ′ ) to complex numbers exists. So starting with X one has arrived at an abelian variety B over W ( k ) and an automorphism σ ′ : B → B offinite order, with possibly finitely many fixed points. Replacing B by a subabelian varietyif needed, one may assume that σ ′ is not a translation on any subvariety of B .Now I proceed by an algebraic variant of [Birkenhake and Lange, 2004, Proposition 13.2.5and Theorem 13.3.2]. This is done as follows. Let Φ d ( X ) be the d –cyclotomic polynomial.So Φ d ( X ) | ( X d −
1) and Φ d ( X ) is irreducible and the primitive d -th roots of unity are itsonly roots. Let f be the endomorphism σ ′ d − d ( σ ′ ) of B , i.e., consider the polynomial f ( X ) = X d − d ( X ) ∈ Z [ X ]and consider the endomorphism f := f ( σ ′ ) : B → B . Consider the subvariety B = f ( B ) ⊂ B . Then B is an abelian variety annihilated by Φ d ( σ ′ ) and hence is naturally a Z [ ζ d ]-module.Moreover B has good ordinary reduction at p , denoted A , and in particular H dR ( B /W ) = H cris ( A /W ) is a Z [ ζ d ] ⊗ Z p W ( k )-module which is finitely generated and Z [ ζ d ]-torsion freeand hence projective of rank k = 2 dim( B ) /φ ( d ). Now every finitely generated projectivemodule over Z [ ζ d ] of rank k is a direct sum of ideals I ⊕ I ⊕ · · · ⊕ I k of Z [ ζ d ]. Using this onesees that, up to isogeny, one may factor B into product of k abelian varieties B , , . . . , B ,k each of dimension φ ( d ) / C this is proved by an analytic argument, attributed to anunpublished result of S. Roan, in [Birkenhake and Lange, 2004, Theorem 13.2.5]). Each ofthese varieties has (possibly up to isogeny) Z [ ζ d ] ֒ → End(B , i ) and as 2 dim( B ,i ) = φ ( d ), soeach has complex multiplication by Z [ ζ d ]. Fix one of these abelian varieties, say, B , . Thenby a basic result [Lang, 1983, Theorem 3.1, page 8] B , is isotypic with a simple abelian variety factor B with complex multiplication by a CM subfield of Q ( ζ d ). Further B has goodordinary reduction at p (by virtue of its construction from B , which has ordinary reductionat p ).On the other hand note that p is totally ramified in the cyclotomic field Q ( ζ d ) as d = p m ≥
3, so p is also totally ramified in the CM subfield for B . Hence one sees, by [Yu, 2003]or [Chai et al., 2014, Prop. 3.7.1.6, Prop. 4.2.6], that the special fiber of B at p is isoclinicof positive slope (equal to half). So it cannot be ordinary. This is a contradiction.Thus d = p m ≤ X is not an abelian variety then one is in case (2). This completesthe proof. (cid:3) If A is an abelian variety then A acts on itself by translations. In particular translation bya non-trivial point of order p is an automorphism of A of order p . In what follows I say thatan automorphism ρ : A → A is a non-trivial automorphism if ρ is not a pure translation.Before proceeding let me point out the following variant of [Igusa, 1955]. Proposition 5.3.
For every algebraically closed field k of characteristic p = 2 or p = 3 ,and for every n ≥ and for every integer N > n , there exists a smooth, projective variety
X/k , of dim( X ) = N , with trivial tangent bundle and a minimal Galois ´etale cover with G = ( Z /p ) n .Proof. Let
A, A , A , . . . , A n be abelian varieties over k satisfying the following conditions:(1) Let ρ i : A i → A i , for 1 ≤ i ≤ n , be a non-trivial automorphism of order p , such thatfor every i the subspace of ρ i -invariant one forms H ( A i , Ω A i ) h ρ i i = H ( A i , Ω A i ),(2) and one has dim( A ) + dim( A ) + · · · + dim( A n ) = N ;(3) suppose A has p -rank at least one.For p = 2 any abelian varieties A, A , . . . , A n satisfying the last two conditions satisfy thefirst with the automorphism ρ i : A i → A i being ρ i ( x ) = − x for all x ∈ A i for 1 ≤ i ≤ n .The condition on invariant forms is trivially satisfied as − p = 2.For p = 3 consider an elliptic curve E/k with a non-trivial automorphism of order p = 3.Let A i = E for 1 ≤ i ≤ n . The condition on invariants is trivially satisfied as Z /p = Z / H ( E, Ω E ) and as any unipotent action has a non-zero subspaceof invariants and as H ( E, Ω E ) is one dimensional, all one forms are invariant under thisnon-trivial automorphism of order three.Thus for any p = 2 , t ∈ A [ p ] with t = 0 be a point on A of order p . Let G = ( Z /p ) n and consider its elements asvectors ( g, g , . . . , g n − ) with entries in Z /p and let G operate on B = A × A × A × · · · × A n as follows: (1 , g , . . . , g n ) · ( x, x , . . . , x n ) = ( x + t, ρ ( x ) , ρ g ( x ) , . . . , ρ g n n ( x n )) , and with the usual convention ρ i = 1 (note the asymmetry in my notation and construction–this is intended to include Igusa surfaces for n = 1 , N = 2). Then G acts free of fixed pointsand the quotient X = B/G is a smooth, projective variety with trivial tangent bundle withminimal ´etale cover with Galois group G and dim( X ) = N . (cid:3) Remark 5.4.
Let me give an example of an abelian variety A in characteristic p > A ) > ρ : A → A of order p , which shows that thecondition on space of invariants is not satisfied in general. Let A be the Jacobian of the hyperelliptic curve y = x p − x . Then the automorphism ( x, y ) ( x + 1 , y ) of y = x p − x isan automorphism of order p (and hence of A ). Using a standard basis for computing forms,one checks that the subspace of invariant forms is not of dimension equal to dim( A ).6. Variants of Li’s Conjecture
In [Li, 2010, Conjecture 4.1] it was conjectured that for p > p > except for the fact that I do not know how to constructabelian varieties satisfying the hypothesis on invariant forms in condition (1) above. But itis possible that abelian varieties satisfying conditions (1)–(3) in the proof of Proposition 5.3might exist for sufficiently large p . Hence in the light of this remark and Theorem 2.4 itseems to me that perhaps Conjecture of [Li, 2010, Conjecture 4.1] needs to be modified. Infact, I propose two separate conjectures, depending on whether one fixes the characteristic orone fixes the dimension. Both the conjectures should be true. The fixed dimension versionis inspired by [Liedtke, 2009]. I note that Conjecture 6.1 replaces [Li, 2010, Conjecture 4.1]. Conjecture 6.1 (Fixed Dimension Version) . Let d ≥ n ( d ) ≥ X/k ofdimension d over an algebraically closed field k and with trivial tangent bundle is an abelianvariety if p > n ( d ). For d = 1, n ( d ) = 1; for d = 2, one has n ( d ) = 3 (by Theorem 3.1).Before I state the fixed characteristic version, let us make the following elementary obser-vation. Lemma 6.2.
There exists an integer n ( p ) ≥ with the following property. For any smooth,projective variety X of dimension d over an algebraically closed field of characteristic p > .If d < n ( p ) then X is an abelian variety.Proof. Suppose, for a given p , there exists a smooth, projective variety Z with trivial tangentbundle which is not an abelian variety. Then for every integer n ≥ dim( Z ), there exists avariety Y of this sort with dim( Y ) = n . Indeed one may simply take Y = Z × E n − dim( Z ) forany elliptic curve E . So take a variety Z with the above properties of the smallest dimensionand let n ( p ) = dim( Z ). If no such variety Z exists one can simply take n ( p ) = 0. Thenevery smooth projective variety X of dimension dim( X ) < n ( p ) is an abelian variety byconstruction. (cid:3) For p = 2 , n ( p ) = 2 by Theorem 3.1. The following is the fixed characteristic versionof the conjecture. Conjecture 6.3 (Fixed Characteristic Version) . Let p be any fixed prime number. Thenumber n ( p ) constructed in Lemma 6.2 has the property that n ( p ) ≥ p ≥ References
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