On a conjecture of Simpson

Abstract

On a compact Riemann surface $\hat{I}\pounds$ of genus $g\ge 2$, equipped with a complex vector bundle $E$ of rank $2$ and degree zero let $M_H$ be the moduli space of Higgs bundles. $M_H$ admits a ${\mathbb{C}}^{\star }$-action and to each stable ${\mathbb{C}}^{\star }$-fixed point $[({\bar{\partial }}_0,\hat{I}{¦}_0)]$ is associated a holomorphic Lagrangian submanifold $W^1({\bar{\partial }}_0,\hat{I}{¦}_0)$ inside the de Rham moduli space $M_{dR}$ of complex flat connections. In this note we prove a conjecture of Simpson stating that $W^1({\bar{\partial }}_0,\hat{I}{¦}_0)$ is closed inside $M_{dR}$.

The proof of the main theorem is new. The proof in the previous version contained a serious mistake found by an anonymous referee. Any comments are welcome