Torelli theorem for the moduli space of symplectic parabolic Higgs bundles

Abstract

Let $(X,D)$ and $(X',D')$ be two compact Riemann surfaces of genus $g \geq 4$ with the set of marked points $D \subset X$ and $D' \subset X'$. Fix a parabolic line bundle $L$ with trivial parabolic structure. Let $\mathcal{N}_{\textnormal{Sp}}(2m,\alpha,L)$ and $\mathcal{N}'_{\textnormal{Sp}}(2m,\alpha,L)$ be the moduli spaces of stable symplectic parabolic Higgs bundles over $X$ and $X'$ respectively, with rank $2m$ and fixed parabolic structure $\alpha$, with the symplectic form taking values in $L$. We prove that if $\mathcal{N}_{\textnormal{Sp}}(2m,\alpha,L)$ is isomorphic to $\mathcal{N}'_{\textnormal{Sp}}(2m,\alpha,L)$, then there exist an isomorphism between $X$ and $X'$ sending $D$ to $D'$.