Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a conjecture of Simpson

Panagiotis Dimakis, Sebastian Schulz

Abstract

On a compact Riemann surface $Σ$ 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 $[(\bar\partial_0,Φ_0)]$ 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 $M_{dR}$.

On a conjecture of Simpson

Abstract

On a compact Riemann surface of genus , equipped with a complex vector bundle of rank and degree zero let be the moduli space of Higgs bundles. admits a -action and to each stable -fixed point is associated a holomorphic Lagrangian submanifold inside the de Rham moduli space of complex flat connections. In this note we prove a conjecture of Simpson stating that is closed inside .

Paper Structure

This paper contains 8 sections, 6 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Background
  4. Deligne's $\lambda$-connections
  5. Holomorphic Lagrangians and the conformal limit
  6. Proof of Simpson's conjecture
  7. WKB and nilpotent Higgs fields
  8. The main theorem

Key Result

Theorem 1.1

Assume that the rank of the bundle $E$ is $2$. Then for any stable $[(\overline\partial_0, \Phi_0)]\in M_H^{\mathbb C^{\star}}$, the holomorphic Lagrangian $W^1(\overline\partial_0, \Phi_0)\subseteq M_{dR}$ is a closed submanifold.

Theorems & Definitions (11)

  • Theorem 1.1: Main Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.1: Proposition $2.7$ in CW
  • Theorem 2.2: Theorem 1.1 in CW
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • ...and 1 more