Related papers: Theta divisors whose Gauss map has a fiber of posi…

We study the degree of the Gauss map of the theta divisor of principally polarised complex abelian varieties. We use this to obtain a bound on the multiplicity of the theta divisor along irreducible components of its singular locus, and…

In this paper we give a lower bound for the codimension of the Andreotti-Mayer loci in the moduli space of principally polarized complex abelian varieties. We also present a conjecture on this codimension.

We study the birational geometry of irregular varieties and the singularities of Theta divisors of PPAV's in positive characteristic by applying recent generic vanishing results of Hacon and Patakfalvi. In particular, we prove that…

We study ample divisors X with only rational singularities on abelian varieties that decompose into a sum of two lower dimensional subvarieties, X=V+W. For instance, we prove an optimal lower bound on the degree of the corresponding…

We survey the geometry of the theta divisor and discuss various loci of principally polarized abelian varieties (ppav) defined by imposing conditions on its singularities. The loci defined in this way include the (generalized)…

We study the loci of principally polarized abelian varieties with points of high multiplicity on the theta divisor. Using the heat equation and degeneration techniques, we relate these loci and their closures to each other, as well as to…

Let $(A,\Theta)$ be a complex principally polarized abelian variety of dimension $g\geq 4$. Based on vanishing theorems, differentiation techniques and intersection theory, we show that whenever the theta divisor $\Theta$ is irreducible,…

In this note we study the geometry of principally polarized abelian varieties (ppavs) with a vanishing theta-null (i.e. with a singular point of order two and even multiplicity lying on the theta divisor). We describe the locus within the…

We use the gradients of theta functions at odd two-torsion points --- thought of as vector-valued modular forms --- to construct holomorphic differential forms on the moduli space of principally polarized abelian varieties, and to…

We use Lagrangian specialization to compute the degree of the Gauss map on Theta divisors with transversal $\mathrm{A}_1$ singularities. This computes the Gauss degree for a general abelian variety in the loci $\mathcal{A}^\delta_{t,g-t}$…

We show that the degree of Gauss maps on abelian varieties is semicontinuous in families, and we study its jump loci. As an application we obtain that in the case of theta divisors this degree answers the Schottky problem. Our proof…

The purpose of this paper is to show how the generic vanishing theorems of M. Green and the second author can be used to settle several questions and conjectures concerning the geometry of irregular complex projective varieties. First, we…

We prove a lower bound for the codimension of the Andreotti-Mayer locus N_{g,1} and show that the lower bound is reached only for the hyperelliptic locus in genus 4 and the Jacobian locus in genus 5. In relation with the boundary of the…

We completely describe the degree of the Gauss map of the theta divisor of bielliptic Prym varieties. We characterize bielliptic Prym varieties whose Gauss degree is the same as Jacobians. We also construct bielliptic Prym varieties with a…

To any closed subvariety $Y$ of a complex abelian variety one can attach a reductive algebraic group $G$ which is determined by the decomposition of the convolution powers of $Y$ via a certain Tannakian formalism. For a theta divisor $Y$ on…

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 (ppav's) of dimension five, is an…

We study the codimension two locus H in A_g consisting of principally polarized abelian varieties whose theta divisor has a singularity that is not an ordinary double point. We compute the class of H in A_g for every g. For g=4, this turns…

We study products of irreducible theta divisors from two points of view. On the one hand, we characterize them as normal subvarieties of abelian varieties such that a desingularization has holomorphic Euler characteristic 1. On the other…

We first state a condition ensuring that having a birational map onto the image is an open property for families of irreducible normal non uniruled varieties. We give then some criteria to ensure general birationality for a family of…

We study the Albanese image of a compact K\"ahler manifold whose geometric genus is one. We prove that if the Albanese map is not surjective, then the manifold maps surjectively onto an ample divisor in some abelian variety, and in many…