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arXiv · 2306.02699

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

Nicholas Rungi·Andrea Tamburelli
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Abstract

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 $\boldsymbolω_G$, we prove that the pair $(\boldsymbolω_G, \mathbf{I})$ cannot define a Kähler structure on the Hitchin component.

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Nicholas Rungi, Andrea Tamburelli. 2024-03-06. Pseudo-Kähler structure on the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component and Goldman symplectic form. https://arxiv.org/abs/2306.02699

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