On a conjecture of Simpson

TL;DR

This paper proves Simpson's conjecture that certain holomorphic Lagrangian submanifolds associated with fixed points in the Higgs bundle moduli space are closed within the de Rham moduli space of flat connections on a compact Riemann surface.

Contribution

It establishes that the holomorphic Lagrangian submanifolds linked to stable fixed points are closed in the de Rham moduli space, confirming a conjecture of Simpson.

Findings

Proves that $W^1(ar extpartial_0, extPhi_0)$ is closed in $M_{dR}$

Confirms Simpson's conjecture for Higgs bundle fixed points

Enhances understanding of the geometry of moduli spaces of flat connections

Abstract

On a compact Riemann surface $\Sigma$ of genus $g>1$, 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 $[(\bar\partial_0,\Phi_0)]$ is associated a holomorphic Lagrangian submanifold $W^1(\bar\partial_0,\Phi_0)$ inside the de Rham moduli space $M_{dR}$ of complex flat connections. In this note we prove a conjecture of Simpson stating that $W^1(\bar\partial_0,\Phi_0)$ is closed inside $M_{dR}$.