Plurisubharmonicity of the Dirichlet energy and deformations of polarized manifolds

Abstract

We show that if $\{M_t\}_{t\in \Delta}$ is a polarized family of compact K\"ahler manifolds over the open unit disk $\Delta$, if $N$ is a Riemannian manifold of nonpositive complexified sectional curvature, and if $\{\phi_t:M_t\to N\}_{t\in \Delta}$ is a smooth family of pluriharmonic maps, then the Dirichlet energy $E(\phi_t)$ is a subharmonic function of $t\in\Delta$.