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

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-Kähler structure.

The main result has now been absorbed into a much larger work currently submitted to arXiv. As well, this paper cites and makes use of content from a previous version of the paper arXiv:2406.15287 that has since been removed. We plan to rewrite this paper and purpose it as an expository note