The behavior of sequences of solutions to the Hitchin-Simpson equations

TL;DR

This paper investigates the limiting behavior of solutions to the Hitchin-Simpson equations on Kähler manifolds, establishing compactness results and applications to harmonic forms and the Hitchin WKB problem.

Contribution

It provides a generalized compactness theorem for solutions with unbounded Higgs fields and applies it to deform harmonic forms and solve the Hitchin WKB problem.

Findings

Proved a compactness result for solutions with unbounded Higgs fields.

Showed every Z/2 harmonic 1-form can be deformed into solutions.

Solved the generalized Hitchin WKB problem on Kähler manifolds.

Abstract

The Hitchin-Simpson equations are first-order non-linear equations for a pair consisting of a connection and a Higgs field. In this paper, we study the behavior of sequences of solutions to the Hitchin-Simpson equations on closed K\"ahler manifolds with unbounded $L^2$ norms of the Higgs fields. We prove a compactness result for the connections and renormalized Higgs fields, which generalizes the work of Taubes and Mochizuki. As applications, we prove that every $\mathbb{Z}/2$ harmonic 1-form on a K\"ahler manifold can be deformed into a sequence of solutions to the Hitchin-Simpson equations. Additionally, we solve the generalized Hitchin's WKB problem on any K\"ahler manifold.