Torelli theorem for the moduli space of symplectic parabolic Higgs bundles

Abstract

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