Compatibility of Goldman's symplectic form with the complex structure on the $\mathrm{SL}(3,\mathbb R)$ Hitchin component

TL;DR

This paper demonstrates that on the $ ext{SL}(3, ext{R})$ Hitchin component, the Goldman symplectic form and the Labourie-Loftin complex structure are compatible, forming a pseudo-Kähler structure invariant under the mapping class group.

Contribution

It establishes the compatibility of symplectic and complex structures on the $ ext{SL}(3, ext{R})$ Hitchin component, revealing a pseudo-Kähler structure.

Findings

Goldman symplectic form and Labourie-Loftin complex structure are compatible

The structures determine a mapping class group invariant pseudo-Kähler structure

The result advances understanding of geometric structures on Hitchin components

Abstract

We prove that, on the $\mathrm{SL}(3,\mathbb R)$ Hitchin component, the Goldman symplectic form and the Labourie-Loftin complex structure are compatible and together determine a (mapping class group invariant) pseudo-K\"ahler structure.