Structure of the Kuranishi Spaces of pairs of K\"ahler manifolds and Polystable Higgs bundles

TL;DR

This paper investigates the local deformation space (Kuranishi space) of pairs consisting of a compact K"ahler manifold and a polystable Higgs bundle with zero Chern classes, showing it decomposes as a product and is unobstructed in certain cases.

Contribution

It establishes an isomorphism of the Kuranishi space of the pair with the product of individual Kuranishi spaces under harmonic metric assumptions, and computes dimensions in special cases.

Findings

Kuranishi space of the pair decomposes as a product.

Deformations are unobstructed for stable Higgs bundles on Riemann surfaces.

Dimension formulas for the Kuranishi space in specific settings.

Abstract

Let $X$ be a compact K\"ahler manifold and $(E,\overline\partial_E,\theta)$ be a Higgs bundle over it. We study the structure of the Kuranishi space for the pair $(X, E,\theta)$ when the Higgs bundle admits a harmonic metric or equivalently when the Higgs bundle is polystable and the Chern classes are 0. Under such assumptions, we show that the Kuranishi space of the pair $(X,E,\theta)$ is isomorphic to the direct product of the Kuranishi space of $(E,\theta)$ and the Kuranishi space of $X$. Moreover, when $X$ is a Riemann surface and $(E,\overline\partial_E,\theta)$ is stable and the degree is 0, we show that the deformation of the pair $(X,E,\theta)$ is unobstructed and calculate the dimension of the Kuranishi space.