On a conjecture of Simpson

Abstract

On a compact Riemann surface $Σ$ 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 $[(\bar\partial_0,Φ_0)]$ is associated a holomorphic Lagrangian submanifold $W^1(\bar\partial_0,Φ_0)$ inside the de Rham moduli space $M_{dR}$ of complex flat connections. In this note we prove a conjecture of Simpson stating that $W^1(\bar\partial_0,Φ_0)$ is closed inside $M_{dR}$.