Generic vanishing and minimal cohomology classes on abelian varieties

We establish a—and conjecture further—relationship between the existence of subvarieties representing minimal cohomology classes on principally polarized abelian varieties, and the generic vanishing of the cohomology of twisted ideal sheaves. The main ingredient is the Generic Vanishing criterion established in Pareschi G. and Popa M. (GV-sheaves, Fourier–Mukai transform, and Generic Vanishing. Preprint math.AG/0608127), based on the Fourier–Mukai transform. MP was partially supported by the NSF grant DMS 0500985 and by an AMS Centennial Fellowship.     

On Ueno’s conjecture K

We show that if X is a smooth complex projective variety with Kodaira dimension 0 then the Kodaira dimension of a general fiber of its Albanese map is at most . J. A. Chen was partially supported by NCTS, TIMS, and NSC of Taiwan. C. D. Hacon was partially supported by NSF research grant no: 0456363 and an AMS Centennial Scholarship. We would like to thank J. Kollár, R. Lazarsfeld, C.-H. Liu, M. Popa, P. Roberts, and A. Singh for many useful comments on the contents of this paper.     

Characteristic classes of Kuga fiber varieties of quaternion type

A Kuga fiber variety is a fiber variety over an arithmetic variety whose fibers are isomorphic to a polarized abelian variety. We determine the Chern classes of Kuga fiber varieties associated to quaternion algebras.     

Densité de points et minoration de hauteur

We obtain a lower bound for the normalised height of a non-torsion subvariety V of a C.M. abelian variety. This lower bound is optimal in terms of the geometric degree of V, up to a power of a “log”. We thus extend the results of Amoroso and David on the same problem on a multiplicative group . We prove furthermore that the optimal lower bound (conjectured by David and Philippon) is a corollary of the conjecture of David and Hindry on the abelian Lehmer's problem. We deduce these results from a density theorem on the non-torsion points of V.     

Rational points on some PEL-stacks

This work studies polarized abelian varieties with endomorphism ring a maximal order in a definite quaternion algebra. The result is that over small fields of characteristic 0 the degree of the polarization can only attain certain values. Received: 2 Febrauary 1999     

Cubic threefolds and abelian varieties of dimension five. II

This paper extends joint work with R. Friedman to show that the closure of the locus of intermediate Jacobians of smooth cubic threefolds, in the moduli space of principally polarized abelian varieties (ppavs) of dimension five, is an irreducible component of the locus of ppavs whose theta divisor has a point of multiplicity three or more. This paper also gives a sharp bound on the multiplicity of a point on the theta divisor of an indecomposable ppav of dimension less than or equal to 5; for dimensions four and five, this improves the bound due to J. Kollár, R. Smith-R. Varley, and L. Ein-R. Lazarsfeld. The author was partially supported by NSF MSPRF grant DMS-0503228.     

Severi varieties

R. Hartshorne conjectured and F. Zak proved (cf [6,p.9]) that any smooth non-degenerate complex algebraic variety with satisfies denotes the secant variety of X; when X is smooth it is simply the union of all the secant and tangent lines to X). In this article, I deal with the limiting case of this theorem, namely the Severi varieties, defined by the conditions and . I want to give a different proof of a theorem of F. Zak classifying all Severi varieties. F. Zak proves that there exists only four Severi varieties and then realises a posteriori that all of them are homogeneous; here I will work in another direction: I prove a priori that any Severi variety is homogeneous and then deduce more quickly their classification, satisfying R. Lazarsfeld et A. Van de Ven's wish [6, p.18]. By the way, I give a very brief proof of the fact that the derivatives of the equation of Sec(X), which is a cubic hypersurface, determine a birational morphism of . I wish to thank Laurent Manivel for helping me in writing this article. Received in final form: 29 March 2001 / Published online: 1 February 2002     

Fixed-point free automorphisms of abelian varieties

An automorphismf of an abelian varietyX is called fixed point free if it admits no fixed points other than the origin and this is of multiplicity one. It is well known that the elliptic curve withj-invariant 0 is the only elliptic curve admitting a fixed point free automorphism. In this note, this result is extended to abelian varieties of higher dimensions and some connected commutative algebraic groups.Supported by DFG-contract La 318/4 and EC-contract SC1-0398-C(A).     

A criterion for an abelian variety to be simple

In this note we give a numerical criterion that expresses the condition that an abelian variety be simple in terms of an invariant that is closely related to the s-invariant of Ein-Cutkosky-Lazarsfeld. Received: 1 July 2007     

Nef cone volumes and discriminants of abelian surfaces

The nef cone volume appeared first in work of Peyre in a number-theoretic context on Fano varieties, and was then studied by Derenthal and co-authors in a series of papers on del Pezzo surfaces. The idea was subsequently extended to also measure the Zariski chambers of del Pezzo surfaces. We start in this paper to explore the possibility to use this attractive concept to effectively measure the size of the nef cone on algebraic surfaces in general. This provides an interesting way of measuring in how big a space an ample line bundle can be moved without destroying its positivity. We give here complete results for simple abelian surfaces that admit a principal polarization and for products of elliptic curves.     

GV‐subschemes and their embeddings in principally polarized abelian varieties

We prove that any embedding of a ‐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 to the Albanese variety of its Fano surface of lines.     

Universal abelian covers of certain surface singularities

Every normal complex surface singularity with -homology sphere link has a universal abelian cover. It has been conjectured by Neumann and Wahl that the universal abelian cover of a rational or minimally elliptic singularity is a complete intersection singularity defined by a system of ``splice diagram equations'. In this paper we introduce a Neumann-Wahl system, which is an analogue of the system of splice diagram equations, and prove the following. If (X, o) is a rational or minimally elliptic singularity, then its universal abelian cover (Y, o) is an equisingular deformation of an isolated complete intersection singularity (Y₀, o) defined by a Neumann-Wahl system. Furthermore, if G denotes the Galois group of the covering Y → X, then G also acts on Y₀ and X is an equisingular deformation of the quotient Y₀/G. Dedicated to Professor Jonathan Wahl on his sixtieth birthday. This research was partially supported by the Grant-in-Aid for Young Scientists (B), The Ministry of Education, Culture, Sports, Science and Technology, Japan.     

Some results about the projective normality of abelian varieties

We reduce the problem of the projective normality of polarized abelian varieties to check the rank of very explicit matrices. This allows us to prove some results on normal generation of primitive line bundles on abelian threefolds and fourfolds. We also give two situations where the projective normality always fails. Finally we make some conjecture. Received: 1 September 2004; revised: 10 March 2005     

Abelian surfaces in projective toric 4-folds

We show fundamental properties on embedding of abelian surfaces into projective toric 4-folds, and study the case of the toric Del Pezzo 4-fold from the viewpoint of the moduli space of abelian surfaces with polarization of type (1, 5). Received: 31 January 2005; revised: 15 April 2005     

A conjecture of Beauville and Catanese revisited

A theorem of Green, Lazarsfeld and Simpson (formerly a conjecture of Beauville and Catanese) states that certain naturally defined subvarieties of the Picard variety of a smooth projective complex variety are unions of translates of abelian subvarieties by torsion points. Their proof uses analytic methods. We refine and give a completely new proof of their result. Our proof combines galois-theoretic methods and algebraic geometry in positive characteristic. When the variety has a model over a function field and its Picard variety has no isotrivial factors, we show how to replace the galois-theoretic results we need by results from model theory (mathematical logic). Furthermore, we prove partial analogs of the conjecture of Beauville and Catanese in positive characteristic.     

Projectively simple rings

An infinite-dimensional N-graded k-algebra A is called projectively simple if dim_kA/I<∞ for every nonzero two-sided ideal I⊂A. We show that if a projectively simple ring A is strongly noetherian, is generated in degree 1, and has a point module, then A is equal in large degree to a twisted homogeneous coordinate ring B=B(X,L,σ). Here X is a smooth projective variety, σ is an automorphism of X with no proper σ-invariant subvariety (we call such automorphisms wild), and L is a σ-ample line bundle. We conjecture that if X admits a wild automorphism then every irreducible component of X is an abelian variety. We prove several results in support of this conjecture; in particular, we show that the conjecture is true if . In the case where X is an abelian variety, we describe all wild automorphisms of X . Finally, we show that if A is projectively simple and admits a balanced dualizing complex, then is Cohen-Macaulay and Gorenstein.     

Projective normality of abelian surfaces¶given by primitive line bundles

In this paper, we study projective normality of abelian surfaces, with embeddings given by ample line bundles of type (1,d). We show that if d≥ 7, the generic abelian surface is projectively normal. Received: 12 June 1998