The Griffiths group $\Gr^r(X)$ of a smooth projective variety $X$ over an algebraically closed field is defined to be the group of homologically trivial algebraic cycles of codimension $r$ on $X$ modulo the subgroup of algebraically trivial algebraic cycles. The main result of this paper is that the Griffiths group $\Gr^2(A_\kbar)$ of a supersingular abelian variety $A_\kbar$ over the algebraic closure of a finite field of characteristic $p$ is at most a $p$-primary torsion group. As a corollary the same conclusion holds for supersingular Fermat threefolds. In contrast, using methods of C. Schoen it is also shown that if the Tate conjecture is valid for all smooth projective surfaces and all finite extensions of the finite ground field $k$ of characteristic $p>2$, then the Griffiths group of any ordinary abelian threefold $A_\kbar$ over the algebraic closure of $k$ is non-trivial.
In this article, we give a complete description of the characteristic polynomials of supersingular abelian varieties over finite fields. We list them for the dimensions upto 7.
In this article, we derive the list of the characteristic polynomials of the Frobenius endomorphism of simple supersingular abelian varieties of dimension $1,~2,~3,~4,~5,~6,~7$ over $\mathbb{F}_q$ where $q=p^n$, $n$ odd.
We study symplectic varieties defined over fields of positive characteristics, especially the supersingular ones, generalizing the theory of supersingular K3 surfaces. In this work, we are mainly interested in the following two types of symplectic varieties over an algebraically closed field of positive characteristic, under natural numerical conditions: (1) smooth moduli spaces of sheaves on K3 surfaces and (2) smooth Albanese fibers of moduli spaces of sheaves on abelian surfaces. Several natural definitions of the supersingularity for symplectic varieties are discussed, which are proved to …
In this article, we derive a list of possible characteristic polynomials of the Frobenius of simple supersingular abelian varieties of dimension 1, 2, 3, 4, 5, 6, 7 over $\mathbb{F}_q$ where $q = p^n$, n odd.
O'Grady constructed a 6-dimensional irreducible holomorphic symplectic variety by taking a crepant resolution of some moduli space of stable sheaves on an abelian surface. In this paper, we naturally extend O'Grady's construction to fields of positive characteristic p greater than 2, called OG6 varieties. We show that a supersingular OG6 variety is unirational, its rational cohomology group is generated by algebraic classes, and its rational Chow motive is of Tate type. These results confirm in this case the generalized Artin--Shioda conjecture, the supersingular Tate conjecture and the supers…
We show that Hilbert schemes of points on supersingular Enriques surface in characteristic 2 are simply connected, symplectic varieties but are not irreducible symplectic as the hodge number $h^{2,0} > 1$, even though a supersingular Enriques surface is an irreducible symplectic variety. These are the classes of varieties which appear only in characteristic 2 and they show that the hodge number formula for Göttsche-Soergel does not hold over characteristic 2. It also gives examples of varieties with trivial canonical class which are neither irreducible symplectic nor Calabi-Yau, thereby showin…
We prove that torsion codimension 2 algebraic cycles modulo rational equivalence on supersingular abelian varieties are algebraically equivalent to zero. As a consequence, we prove that homological equivalence coincides with algebraic equivalence for algebraic cycles of codimension 2 on supersingular abelian varieties over the algebraic closure of finite fields.
By the fundamental work of Griffiths one knows that, under suitable assumption, homological and algebraic equivalence do not coincide for a general hypersurface section of a smooth projective variety $Y$. In the present paper we prove the same result in case $Y$ has isolated singularities.
In this paper, we use a group-theoretic approach to give a concrete description of the geometric structure of the supersingular locus of unitary Shimura varieties with exotic good reduction. This approach also is a more uniform way to prove results of this form obtained previously by, for example, Vollaard-Wedhorn and Rapoport-Terstiege-Wilson.
An abelian variety defined over an algebraically closed field k of positive characteristic is supersingular if it is isogenous to a product of supersingular elliptic curves and is superspecial if it is isomorphic to a product of supersingular elliptic curves. In this paper, the superspecial condition is generalized by defining the superspecial rank of an abelian variety, which is an invariant of its p-torsion. The main results in this paper are about the superspecial rank of supersingular abelian varieties and Jacobians of curves. For example, it turns out that the superspecial rank determines…
In this note, we study Shimura varieties for the groups $\mathrm{GU}(V)$, where $V$ is a Hermitian space relative to a CM extension $E/E^+$. We give a description of the supersingular locus of the fiber at a prime $ν$ over $p$ of such a Shimura variety, under the assumptions that $\mathrm{dim}_E(V) \leq 4$ and that the prime $p$ splits completely in $E^+$ and is unramified in $E$.
We investigate Siegel modular varieties in positive characteristic with Iwahori level structure. On these spaces, we have the Newton stratification, and the Kottwitz-Rapoport stratification; one would like to understand how these stratifications are related to each other. We give a simple description of all KR strata which are entirely contained in the supersingular locus as disjoint unions of Deligne-Lusztig varieties. We also give an explicit numerical description of the KR stratification in terms of abelian varieties.
We investigate to what extent an abelian variety over a finite field can be lifted to one in characteristic zero with small Mumford-Tate group. We prove that supersingular abelian surfaces, respectively threefolds, can be lifted to ones isogenous to a square, respectively product, of elliptic curves. On the other hand, we show that supersingular abelian threefolds cannot be lifted to one isogenous to the cube of an elliptic curve over the Witt vectors.
We introduce and study a new way to catagorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed, or fully minimal. The type of $A$ depends on the normalized Weil numbers of $A$ and its twists. We analyze these types for supersingular abelian varieties and curves under conditions on the automorphism group. In particular, we present a complete analysis of these properties for supersingular elliptic curves and supersingular abelian surfaces in arbitrary characteristic, and for a one-dimensional family of supersingular curves of genus $3$ in c…
We study moduli spaces of abelian varieties in positive characteristic, more specifically the moduli space of principally polarized abelian varieties on the one hand, and the analogous space with Iwahori type level structure, on the other hand. We investigate the Ekedahl-Oort stratification on the former, the Kottwitz-Rapoport stratification on the latter, and their relationship. In this way, we obtain structural results about the supersingular locus in the case of Iwahori level structure, for instance a formula for its dimension in case $g$ is even.
This article has three goals. First, we generalize the result of Deuring and Serre on the characterization of supersingular locus of modular curves to all Shimura varieties given by totally indefinite quaternion algebras over totally real number fields. Second, we generalize the result of Ribet on arithmetic level raising to such Shimura varieties in the inert case. Third, as an application to number theory, we use the previous results to study the Selmer group of certain triple product motive of an elliptic curve, in the context of the Bloch--Kato conjecture.
We give a short proof of the "prime-to-$p$ version" of the Manin-Mumford conjecture for an abelian variety over a number field, when it has supersingular reduction at a prime dividing $p$, by combining the methods of Bogomolov, Hrushovski, and Pink-Roessler. Our proof here is quite simple and short, and neither $p$-adic Hodge theory nor model theory is used. The observation is that a power of a lift of the Frobenius element at a supersingular prime acts on the prime-to-$p$ torsion points via nontrivial homothety.
We exhibit examples of projective varieties with degenerate Gauss mappings and determine numerical invariants of such varieties. Our examples provide counter-examples to an asserted structure theorem of Griffiths and Harris (Ann. Sci. ENS 1979).
Upper bounds on projective rigidity of each homogeneously embedded homogeneous variety are determined; and a new, invariant characterization of the Fubini forms is given.
We study the moduli space A_g of g-dimensional principally polarized abelian varieties in positive characteristic, and its variant A_I with Iwahori level structure. Both supersingular Ekedahl-Oort strata and supersingular Kottwitz-Rapoport strata are isomorphic to disjoint unions of Deligne-Lusztig varieties (see [Hoeve 2008] and [Goertz, Yu 2008], resp.). Here we compare these isomorphisms. We also give an explicit description of Kottwitz-Rapoport strata contained in the supersingular locus in the general parahoric case. Finally, we show that every Ekedahl-Oort stratum is isomorphic to a para…
We show that there is a good notion of irreducible sympelectic varieties of $\mathrm{K3}^{[n]}$-type over an arbitrary field of characteristic zero or $p > n + 1$. Then we construct mixed characteristic moduli spaces for these varieties. Our main result is a generalization of Ogus' crystalline Torelli theorem for supersingular K3 surfaces. For applications, we answer a slight variant of a question asked by F. Charles on moduli spaces of sheaves on K3 surfaces and give a crystalline Torelli theorem for supersingular cubic fourfolds.
Under the axiom of He and Rapoport for the stratifications of Shimura varieties, we explain a result of Görtz, He and Nie that the EKOR strata contained in the basic loci can be described as a disjoint union of Deligne-Lusztig varieties. In the special case of Siegel modular varieties, we compare their descriptions to that of Görtz and Yu for the supersingular Kottwitz-Rapoport strata and to the descriptions of Harashita, Hoeve for the supersingular Ekedahl-Oort strata.
We establish vanishing results for spaces of automorphic forms in characteristic $0$ and characteristic $p$. We prove that for Hodge-type Shimura varieties, the weight of any nonzero automorphic form in characteristic $0$ satisfies the Griffiths-Schmid conditions, by purely algebraic, characteristic $p$ methods. We state a conjecture for general Hodge-type Shimura varieties regarding the vanishing of the space of automorphic forms in characteristic $p$ in terms of the weight. We verify this conjecture for unitary PEL Shimura varieties of signature $(n-1,1)$ at a split prime.
We describe the structure of the supersingular locus of a Shimura variety for a quaternionic unitary similitude group of degree $2$ over a ramified odd prime $p$ if the level at $p$ is given by a special maximal compact open subgroup. More precisely, we show that such a locus has purely $2$-dimensional, and every irreducible component is birational to the Fermat surface. Furthermore, we have an estimation of the numbers of connected and irreducible components. To prove these assertions, we completely determine the structure of the underlying reduced scheme of the Rapoport--Zink space for the q…
Let L be a quadratic imaginary field, inert at the rational prime p. Fix an integer n at least 3, and let M be the moduli space (in characteristic p) of principally polarized abelian varieties of dimension n equipped with an action by O_L of signature of (1,n-1). We show that each Newton stratum of M, other than the supersingular stratum, is irreducible.
We show that there are smooth complex projective varieties with infinite 2-torsion in their third Griffiths groups. It follows that the torsion subgroup of Griffiths groups is in general not finitely generated, thereby solving a problem of Schoen from 1992.
We define Frobenius incidence varieties by means of the incidence relation of Frobenius images of linear subspaces in a fixed vector space over a finite field, and investigate their properties such as supersingularity, Betti numbers and unirationality. These varieties are variants of the Deligne-Lusztig varieties. We then study the lattices associated with algebraic cycles on them. We obtain a positive-definite lattice of rank 84 that yields a dense sphere packing from a 4-dimensional Frobenius incidence variety in characteristic 2.
We prove a boundedness-theorem for families of abelian varieties with real multiplication. More generally, we study curves in Hilbert modular varieties from the point of view of the Green Griffiths-Lang conjecture claiming that entire curves in complex projective varieties of general type should be contained in a proper subvariety. Using holomorphic foliations theory, we establish a Second Main Theorem following Nevanlinna theory. Finally, with a metric approach, we establish the strong Green-Griffiths-Lang conjecture for Hilbert modular varieties up to finitely many possible exceptions.
There is a well known theorem by Deuring which gives a criterion for when the reduction of an elliptic curve with complex multiplication (CM) by the ring of integers of an imaginary quadratic field has ordinary or supersingular reduction. We generalise this and a similar theorem by Goren in dimension 2, and classify the $p$-torsion group scheme of the reduction of 3-dimensional abelian varieties with CM by the ring of integers of a cyclic sextic CM field. We also prove a theorem in arbitrary dimension $g$ that distinguishes ordinary and superspecial reduction for abelian varieties with CM by a…
By using techniques of holomorphic jets and Jacobian fields, we devise a non-equidistribution theory of holomorphic curves into complex projective varieties intersecting normal crossing divisors. Based on this theory established, we prove the Griffiths conjecture and the Green-Griffiths conjecture in Nevanlinna theory and algebraic geometry.
We prove that the embedding of a $GV$-subscheme in a principally polarized abelian variety does not factor through any nontrivial isogeny. As an application, we present a new proof of a theorem of Clemens--Griffiths identifying the intermediate Jacobian of a smooth cubic threefold in $\mathbf{P}^4$ to the Albanese variety of its Fano surface of lines.
We prove that the Griffiths group of 3-cycles homologous to zero modulo algebraic equivalence, on a generic hypersurfaces of dimension 7 and degree 3 is not finitely generated, even when tensored with Q. Using this and a result of Nori, we give examples of varieties for which some Griffiths group is not finitely generated (modulo torsion) but whose corresponding intermediate Jacobian is trivial.
In this paper, we study the set $R_g^{(p)}$ of possible Picard numbers of abelian varieties of dimension $g$ over algebraically closed fields of characteristic $p>0$. We show that many of the results for complex abelian varieties have analogues in positive characteristic: non-completeness in dimension $g \geq 2$, asymptotic completeness as $g \rightarrow +\infty$, structure results for abelian varieties of large Picard number. On the way, we highlight and discuss new characteristic $p>0$ features and pathologies: non-additivity of the range of Picard numbers, supersingularity index of an abeli…
We prove the congruence relation for the mod-p reduction of Shimura varieties associated to a unitary similitude group GU(n-1,1), when p is inert and n odd. When n is even, this result was obtained by T. Wedhorn and O. Bültel using the density of the ordinary locus in the p-isogeny space. This condition fails in our case. A key element is the understanding of the supersingular locus, for which we refer to the two articles by I. Vollaard and T. Wedhorn. The proof makes extensive use of elementary algebraic geometry, but also some deeper results.
We study some automorphic cohomology classes of degree one on the Griffiths-Schmid varieties attached to some unitary groups in 3 variables. Using partial compactifications of those varieties, constructed by K. Kato and S. Usui, we define for such a cohomology class some analogues of Fourier-Shimura coefficients, which are cohomology classes on certain elliptic curves. We show that a large space of such automorphic classes can be generated by those with rational "coefficients". More precisely, we consider those cohomology classes that come from Picard modular forms, via some Penrose-like trans…
We prove results about 1-cycles on certain Fano varieties using techniques that rely on rational curves. Firstly, we show that Fano weighted complete intersections with index bigger than half their dimension have trivial first Griffiths group. Secondly, we prove that the first Chow group of most $2$-Fano weighted complete intersections, and of $2$-Fano conic-connected varieties in $\mathbb{P}^n$ of high enough index (with $3$ obvious exceptions), are generated by lines. Furthermore, if the Fano variety of lines is irreducible, the first Chow group is isomorphic to $\mathbb{Z}$.
We prove a full generalization of the Castelnuovo's free pencil trick. We show its analogies with the Adjoint Theorem; see L. Rizzi, F. Zucconi, Differential forms and quadrics of the canonical image, arXiv:1409.1826 and also Theorem 1.5.1 in G. P. Pirola, F. Zucconi, Variations of the Albanese morphisms, J. Algebraic Geom. 12 (2003), no. 3, 535-572. Moreover we find a new formulation of the Griffiths's infinitesimal Torelli Theorem for smooth projective hypersurfaces using meromorphic $1$-forms.
We construct a map between Bloch's higher Chow groups and Deligne homology for smooth, complex quasiprojective varieties on the level of complexes. For complex projective varieties this results in a formula which generalizes at the same time the classical Griffiths Abel-Jacobi map and the Borel/Beilinson/Goncharov regulator type maps.
We formulate a conjectural hard Lefschetz property for Chow groups, and prove this in some special cases: roughly speaking, for varieties with finite-dimensional motive, and for varieties whose self-product has vanishing middle-dimensional Griffiths group. An appendix includes related statements that follow from results of Vial.
The study of entire holomorphic curves contained in projective algebraic varieties is intimately related to fascinating questions of geometry and number theory -- especially through the concepts of curvature and positivity which are central themes in Kodaira's contributions to mathematics. The aim of these lectures is to present recent results concerning the geometric side of the problem. The Green-Griffiths-Lang conjecture stipulates that for every projective variety $X$ of general type over~${\mathbb C}$, there exists a proper algebraic subvariety $Y$ of $X$ containing all non constant entir…
Let F(X_d) be a smooth Fano variety of lines of a hypersurface X_d of degree d. In this paper, we prove the Griffiths group Griff_1(F(X_d)) is trivial if the hypersurface X_d is of 2-Fano type. As a result, we give a positive answer to a question of Professor Voisin about the first Griffiths groups of Fano varieties in some cases. Base on this result, we prove that CH_2(X_d)=\mathbb{Z} for a complex smooth $3$-Fano hypersurface X_d whose Fano variety of lines is smooth.
We prove a general extrinsic rigidity theorem for homogeneous varieties in $\mathbb{CP}^N$. The theorem is used to show that the adjoint variety of a complex simple Lie algebra $\mathfrak{g}$ (the unique minimal G orbit in $\mathbb{P}\mathfrak{g}$) is extrinsically rigid to third order. In contrast, we show that the adjoint variety of $SL_3\mathbb{C}$, and the Segre product $\mathit{Seg}(\mathbb{P}^1\times \mathbb{P}^n)$, both varieties with osculating sequences of length two, are flexible at order two. In the $SL_3\mathbb{C}$ example we discuss the relationship between the extrinsic projectiv…
