The Hitchin-cscK system

TL;DR

This paper introduces an extended hyperk"ahler reduction framework that generalizes the scalar curvature equation for K"ahler metrics by incorporating a Higgs field, with applications to complex curves and certain ruled surfaces.

Contribution

It develops a new system of moment map equations extending the cscK condition, inspired by Hitchin's equations, and explores solutions on various complex surfaces.

Findings

Extended existence results for the new system on Riemann surfaces.

Showed that adding a Higgs field can stabilize certain non-cscK ruled surfaces.

Applied symplectic coordinate methods to analyze solutions on abelian and toric surfaces.

Abstract

We present an infinite-dimensional hyperk\"ahler reduction that extends the classical moment map picture of Fujiki and Donaldson for the scalar curvature of K\"ahler metrics. We base our approach on an explicit construction of hyperk\"ahler metrics due to Biquard and Gauduchon. The construction is motivated by how one can derive Hitchin's equations for harmonic bundles from the Hermitian Yang-Mills equation, and yields a system of moment map equations which modifies the constant scalar curvature K\"ahler (cscK) condition by adding a "Higgs field" to the cscK equation. In the special case of complex curves, we recover previous results of Donaldson, while for higher-dimensional manifolds the system of equations has not yet been studied. We study the existence of solutions to the system in some special cases. On a Riemann surface, we extend an existence result for Donaldson's equation to our system. We then study the existence of solutions to the moment map equations on a class of ruled surfaces which do not admit cscK metrics, showing that adding a suitable Higgs term to the cscK equation can stabilize the manifold. Lastly, we study the system of equations on abelian and toric surfaces, taking advantage of a description of the system in symplectic coordinates analogous to Abreu's formula for the scalar curvature.