Existence and unicity of pluriharmonic maps to Euclidean buildings and applications

TL;DR

This paper proves the existence and uniqueness of certain pluriharmonic maps from complex varieties to Euclidean buildings, with applications to geometric group theory and algebraic geometry.

Contribution

It establishes the existence and uniqueness of equivariant pluriharmonic maps into Bruhat-Tits buildings for Zariski dense representations.

Findings

Constructed $ ho$-equivariant pluriharmonic maps with specified asymptotics

Proved the uniqueness of these maps under certain conditions

Provided a geometric characterization of the maps

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.