Pseudo-Kähler structure on the $\mathrm{S}\mathrm{L}(3,\mathbb{R})$-Hitchin component and Goldman symplectic form

The aim of this paper is to show the existence and give an explicit description of a pseudo-Riemannian metric and a symplectic form on the $\mathrm{S}\mathrm{L}(3,\mathbb{R})$-Hitchin component, both compatible with Labourie and Loftin’s complex structure. In particular, they give rise to a mapping class group invariant pseudo-Kähler structure on a neighborhood of the Fuchsian locus, which restricts to a multiple of the Weil-Petersson metric on Teichmüller space. By comparing our symplectic form with Goldman’s ${\mathit{\omega }}_G$, we prove that the pair $({\mathit{\omega }}_G,\mathbf{I})$ cannot define a Kähler structure on the Hitchin component.