A Hodge theoretic projective structure on Riemann surfaces

TL;DR

This paper compares two canonical projective structures on Riemann surfaces—one from a meromorphic 2-form and the other from uniformization—showing they generally differ by analyzing their differentials over moduli space.

Contribution

It establishes that the projective structure from the meromorphic 2-form differs from the uniformization structure, linking the differential to the Siegel form via the Torelli map.

Findings

The differential of the meromorphic 2-form-based structure is the pullback of the Siegel form.

The two projective structures generally do not coincide.

The differential of the uniformization-based structure relates to the Weil-Petersson form.

Abstract

Given any compact Riemann surface $C$, there is a canonical meromorphic 2--form $\widehat\eta$ on $C\times C$, with pole of order two on the diagonal $\Delta\, \subset\, C\times C$, constructed in \cite{cfg}. This meromorphic 2--form $\widehat\eta$ produces a canonical projective structure on $C$. On the other hand the uniformization theorem provides another canonical projective structure on any compact Riemann surface $C$. We prove that these two projective structures differ in general. This is done by comparing the $(0,1)$--component of the differential of the corresponding sections of the moduli space of projective structures over the moduli space of curves. The $(0,1)$--component of the differential of the section corresponding to the projective structure given by the uniformization theorem was computed by Zograf and Takhtadzhyan in \cite{ZT} as the Weil--Petersson K\"ahler form $\omega_{wp}$ on the moduli space of curves. We prove that the $(0,1)$--component of the differential of the section of the moduli space of projective structures corresponding to $\widehat{\eta}$ is the pullback of a nonzero constant scalar multiple of the Siegel form, on the moduli space of principally polarized abelian varieties, by the Torelli map.