Existence and unicity of pluriharmonic maps to Euclidean buildings and applications

Abstract

Given a complex smooth quasi-projective variety $X$, a reductive algebraic group $G$ defined over some non-archimedean local field $K$ and a Zariski dense representation $\varrho:\pi_1(X)\to G(K)$, we construct a $\varrho$-equivariant pluriharmonic map from the universal cover of $X$ into the Bruhat-Tits building $\Delta(G)$ of $G$, with appropriate asymptotic behavior. We also establish the uniqueness of such a pluriharmonic map in a suitable sense, and provide a geometric characterization of these equivariant maps. This paper builds upon and extends previous work by the authors jointly with G. Daskalopoulos and D. Brotbek.