Bergman kernel and period map for curves

TL;DR

This paper explores the relationship between the Bergman kernel, the period map, and the geometry of algebraic curves, providing new insights into the Torelli map's differential properties and harmonic representatives.

Contribution

It characterizes the pull-back operation on the moduli space of curves using the Bergman kernel form, linking it to the second fundamental form of the Torelli map.

Findings

Bergman kernel form is the harmonic representative of a meromorphic form related to the Torelli map.

The second fundamental form of the Torelli map can be described via multiplication by a meromorphic form.

The work provides a geometric interpretation of the tangent space operation on the moduli space of curves.

Abstract

As for any symmetric space the tangent space to Siegel upper-half space is endowed with an operation coming from the Lie bracket on the Lie algebra. We consider the pull-back of this operation to the moduli space of curves via the Torelli map. We characterize it in terms of the geometry of the curve, using the Bergman kernel form associated to the curve. It is known that the second fundamental form of the Torelli map outside the hyperelliptic locus can be seen as the multiplication by a certain meromorphic form. Our second result says that the Bergman kernel form is the harmonic representative - in a suitable sense - of this meromorphic form.