Projective structures and Hodge theory

TL;DR

This paper introduces a new Hodge projective structure on compact Riemann surfaces, compares it with the classical uniformization structure, and explores their relationship via moduli space differentials.

Contribution

It constructs the Hodge projective structure related to the period map's second fundamental form and analyzes its distinction from the uniformization structure.

Findings

The Hodge projective structure is distinct from the uniformization structure.

Projective structures correspond to (1,1)-forms on the moduli space.

The paper clarifies the geometric relationship between different natural projective structures.

Abstract

Every compact Riemann surface $X$ admits a natural projective structure $p_u$ as a consequence of the uniformization theorem. In this work we describe the construction of another natural projective structure on $X$, namely the Hodge projective structure $p_h$, related to the second fundamental form of the period map. We then describe how projective structures correspond to $(1,1)$-differential forms on the moduli space of projective curves and, from this correspondence, we deduce that $p_u$ and $p_h$ are not the same structure.