Prym varieties and projective structures on Riemann surfaces

TL;DR

This paper explores how Prym varieties associated with étale double covers of Riemann surfaces can be used to construct canonical projective structures, revealing their dependence on the covering in the moduli space.

Contribution

It introduces a new construction of projective structures on Riemann surfaces using Prym varieties and analyzes their variation via the $ar{ abla}$-derivative in relation to moduli space geometry.

Findings

Constructs canonical projective structures from Prym varieties.

Computes the $ar{ abla}$-derivative in terms of Thetanullwert maps.

Shows projective structures depend on the cover in general.

Abstract

Given an \'etale double covering $\pi\, :\, \widetilde{C}\, \longrightarrow\, C$ of compact Riemannsurfaces with $C$ of genus at least two, we use the Prym variety of the cover to construct canonical projective structures on both $\widetilde C$ and $C$. This construction can be interpreted as a section of an affine bundle over the moduli space of \'etale double covers. The $\overline{\partial}$--derivative of this section is a (1,1)--form on the moduli space. We compute this derivative in terms of Thetanullwert maps. Using the Schottky--Jung identities we show that, in general, the projective structure on $C$ depends on the cover.