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 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-K & auml;hler structure that restricts to a multiple of the Weil-Petersson metric on Teichm & uuml;ller space. By comparing our symplectic form with Goldman's omega G , we prove that the pair (omega G, I ) cannot define a K & auml;hler structure on the Hitchin component. (c) 2024 Published by Elsevier Inc.