A semi-pseudo-Kähler structure on the SL(3,R)-Hitchin component and the Goldman symplectic form

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