A desingularization of the moduli space of rank 2 Higgs bundles over a curve

Let $X$ be a smooth complex projective curve of genus $g\geq 3$. Let $\mathbf{M}_2$ be the moduli space of semistable rank $2$ Higgs bundles with trivial determinant over $X$. We construct a desingularization $\mathbf{S}$ of this singular moduli space as a closed subvariety of a bigger moduli space. The main purpose of this paper is to prove that $\mathbf{S}$ is a nonsingular variety containing the stable locus of $\mathbf{M}_2$ as an open dense subvariety. On the other hand, there is another desingularization $\mathbf{K}$ of $\mathbf{M}_2$ by Kirwan's algorithm. Following O'Grady's argument, a nonsingular variety $\mathbf{K}_\epsilon$ can be obtained after two blow-downs of $\mathbf{K}$. There is no known information on the moduli theoretic meaning of $\mathbf{K}_\epsilon$. As an application of our main result, we show that $\mathbf{K}_\epsilon$ is isomorphic to $\mathbf{S}$. Using this application, we can also get a conjectural closed formula for Poincar\'e series of $\mathbf{S}$.