Complex structures, moment maps, and the Ricci form (Extended Version)

TL;DR

This paper explores the Ricci form as a moment map for volume-preserving diffeomorphisms acting on almost complex structures, offering a novel perspective on the Weil-Petersson form in Teichmüller theory.

Contribution

It introduces a new geometric interpretation of the Ricci form as a moment map, connecting complex structures, symplectic forms, and diffeomorphism groups.

Findings

Ricci form identified as a moment map for volume-preserving diffeomorphisms.

Provides a new approach to understanding the Weil-Petersson symplectic form.

Establishes links between complex geometry and symplectic structures in Teichmüller space.

Abstract

The Ricci form is a moment map for the action of the group of exact volume preserving diffeomorphisms on the space of almost complex structures. This observation yields a new approach to the Weil-Petersson symplectic form on the Teichmuller space of isotopy classes of complex structures with real first Chern class zero and nonempty Kahler cone.