Restriction of Donaldson's functional to diagonal metrics on Higgs bundles with non-holomorphic Higgs fields

TL;DR

This paper studies the minimization of Donaldson's functional on diagonal metrics for Higgs bundles with non-holomorphic Higgs fields over compact Kähler manifolds, providing conditions for solutions even when the Higgs field is non-holomorphic.

Contribution

It establishes necessary and sufficient conditions for Donaldson's functional to attain a minimum on diagonal metrics in non-holomorphic Higgs bundles, extending solutions to the Hermitian-Einstein equation.

Findings

Conditions for the functional's minimum are derived.

Solutions to the Hermitian-Einstein equation are obtained under strong assumptions.

The results apply even when the Higgs field is non-holomorphic.

Abstract

We consider a Higgs bundle over a compact K\"ahler manifold with a smooth, non-holomorphic Higgs field. We assume that the holomorphic vector bundle decomposes into a direct sum of holomorphic line bundles. Under an assumption on the zero set of the non-holomorphic Higgs field, we provide some necessary and sufficient conditions for Donaldson's functional which is restricted to the set of diagonal Hermitian metrics associated with a holomorphic decomposition of the vector bundle to attain a minimum. In particular, when the holomorphic vector bundle decomposes into a direct sum of holomorphic line bundles, we show that we can solve the Hermitian-Einstein equation under a strong assumption even if the Higgs field is non-holomorphic.