Semicontinuity of Gauss maps and the Schottky problem

TL;DR

This paper investigates the semicontinuity of Gauss map degrees on abelian varieties, linking their behavior to the Schottky problem and providing new insights into the topology of theta divisors within the moduli space.

Contribution

It establishes the semicontinuity of Gauss map degrees and connects these to the Schottky problem, using specialization of Lagrangian cycles and analyzing the topology of theta divisors.

Findings

Gauss map degree is semicontinuous in families of abelian varieties.

The degree of Gauss maps characterizes the Schottky problem for theta divisors.

Components of Andreotti-Mayer loci are stratified by the topological type of theta divisors.

Abstract

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 computes the degree of Gauss maps by specialization of Lagrangian cycles on the cotangent bundle. We also get similar results for the intersection cohomology of varieties with a finite morphism to an abelian variety; it follows that many components of Andreotti-Mayer loci, including the Schottky locus, are part of the stratification of the moduli space of ppav's defined by the topological type of the theta divisor.