Hermitian symmetric space, flat bundle and holomorphicity criterion

TL;DR

This paper establishes criteria for harmonic maps from compact Kähler manifolds to Hermitian symmetric spaces to be holomorphic or anti-holomorphic, based on the properties of associated group representations and harmonic map theory.

Contribution

It provides new conditions under which harmonic maps into Hermitian symmetric spaces are holomorphic or anti-holomorphic, extending previous understanding in differential geometry and representation theory.

Findings

Criteria for harmonic maps to be holomorphic

Criteria for harmonic maps to be anti-holomorphic

Connection between group representations and harmonic map properties

Abstract

Let $K\backslash G$ be an irreducible Hermitian symmetric space of noncompact type and $\Gamma \,\subset\, G$ a closed torsionfree discrete subgroup. Let $X$ be a compact K\"ahler manifold and $\rho\, :\, \pi_1(X, x_0)\,\longrightarrow\, \Gamma$ a homomorphism such that the adjoint action of $\rho(\pi_1(X, x_0))$ on $\text{Lie}(G)$ is completely reducible. A theorem of Corlette associates to $\rho$ a harmonic map $X\, \longrightarrow\, K\backslash G/\Gamma$. We give a criterion for this harmonic map to be holomorphic. We also give a criterion for it to be anti--holomorphic.