Global rigidity of the period mapping

TL;DR

This paper proves that for genus g ≥ 3, any nonconstant holomorphic map from the moduli space of curves to the moduli space of abelian varieties must be the classical period mapping, which assigns Jacobians to Riemann surfaces.

Contribution

It establishes the global rigidity of the period mapping, showing it is uniquely characterized among holomorphic maps between these moduli spaces.

Findings

Any nonconstant holomorphic map from ${ m extbf{M}}_{g,n}$ to ${ m extbf{A}}_h$ with $g eq h$ is constant.

For $g eq h$, such maps do not exist; when $g=h$, the map is the classical period mapping.

The result confirms the uniqueness of the period mapping in the context of moduli spaces.

Abstract

Let ${\mathcal M}_{g,n}$ denote the moduli space of smooth, genus $g\geq 1$ curves with $n\geq 0$ marked points. Let ${\mathcal A}_h$ denote the moduli space of $h$-dimensional, principally polarized abelian varieties. Let $g\geq 3$ and $h\leq g$. If $F:{\mathcal M}_{g,n}\to{\mathcal A}_h$ is a nonconstant holomorphic map then $h=g$ and $F$ is the classical period mapping, assigning to a Riemann surface $X$ its Jacobian.