A Lefschetz Hyperplane Theorem for non-Archimedean Jacobians

TL;DR

This paper proves a Lefschetz hyperplane theorem for non-Archimedean Jacobians' analytifications, revealing topological properties of divisor loci and theta divisors in this setting.

Contribution

It establishes a Lefschetz hyperplane theorem for Berkovich analytifications of Jacobians over non-Archimedean fields, generalizing previous results to arbitrary characteristics and singularities.

Findings

The pair (J^{an},W_d^{an}) is d-connected.

The inclusion of the theta divisor satisfies a Lefschetz theorem for cohomology and homotopy.

Generalization of Brown and Foster's result to arbitrary characteristics and singularities.

Abstract

We establish a Lefschetz hyperplane theorem for the Berkovich analytifications of Jacobians of curves over an algebraically closed non-Archimedean field. Let $J$ be the Jacobian of a curve $X$, and let $W_d \subset J$ be the locus of effective divisor classes of degree $d$. We show that the pair $(J^{an},W_d^{an})$ is $d$-connected, and thus in particular the inclusion of the analytification of the theta divisor $\Theta^{an}$ into $J^{an}$ satisfies a Lefschetz hyperplane theorem for $\mathbb{Z}$-cohomology groups and homotopy groups. A key ingredient in our proof is a generalization, over arbitrary characteristics and allowing arbitrary singularities on the base, of a result of Brown and Foster for the homotopy type of analytic projective bundles.