Infinitesimal Deformations of Generalized Parabolic Hitchin Pairs

Abstract

We develop the infinitesimal deformation theory of generalized parabolic Hitchin pairs (GPHs) on irreducible nodal curves. For any GPH $(E,\varphi ,F(E))$ we associate an explicit three-term deformation complex ${\mathcal{C}}_{(E_{\bullet },\varphi )}$ whose first hypercohomology ${\mathbb{H}}^1({\mathcal{C}}_{(E_{\bullet },\varphi )})$ parameterizes first-order deformations. As the main application, we prove that the coarse moduli space ${\mathcal{M}}_{\mathrm{GPH}}$ of $1$-stable generalized parabolic Hitchin pairs of rank $n$ and degree $d$ with $\gcd (n,d)=1$ is a local complete intersection, smooth in codimension one, and hence a normal variety. We further show that this moduli space carries a natural Poisson structure.