Harmonic maps from K\"ahler manifolds

TL;DR

This paper provides a comprehensive overview of harmonic maps from K"ahler manifolds, highlighting key results, techniques, and applications in complex geometry and nonabelian Hodge theory.

Contribution

It offers a clear presentation of harmonic maps from K"ahler manifolds, including refined Bochner formulas, rigidity results, and their role in nonabelian Hodge correspondence.

Findings

Refined Bochner formula due to Siu and Sampson

Strong rigidity results for harmonic maps from K"ahler manifolds

Applications to nonabelian Hodge theory and character varieties

Abstract

This report attempts a clean presentation of the theory of harmonic maps from complex and K\"ahler manifolds to Riemannian manifolds. After reviewing the theory of harmonic maps between Riemannian manifolds initiated by Eells--Sampson and the Bochner technique, we specialize to K\"ahler domains and introduce pluriharmonic maps. We prove a refined Bochner formula due to Siu and Sampson and its main consequences, such as the strong rigidity results of Siu. We also recount the applications to symmetric spaces of noncompact type and their relation to Mostow rigidity. Finally, we explain the key role of this theory for the nonabelian Hodge correspondence relating the character variety of a compact K\"ahler manifold and the moduli space of Higgs bundles.