Existence and unicity of pluriharmonic maps to Euclidean buildings and applications
Ya Deng, Chikako Mese
TL;DR
The paper proves existence and (local) uniqueness of ρ-equivariant pluriharmonic maps from the universal cover of a smooth quasi-projective variety X into the enlarged Bruhat–Tits building Δ(G), with logarithmic energy growth and compatibility under pullback. It develops a geometric spectral-cover framework: a map induces a multivalued logarithmic 1-form η, which, after passing to a ramified spectral cover and resolving, becomes a collection of pure imaginary, single-valued logarithmic forms; η encodes the eigen-structure of the map and is independent of the chosen pluriharmonic map. The authors decompose the construction into a semisimple part via DG and a vector part via the quasi-Albanese map, producing a pluriharmonic map whose energy and regularity properties are controlled, and they show a strong unicity statement on a dense open set. They further analyze the singular set, proving it lies in a proper Zariski-closed subset, and establish a robust spectral-form description that links differential-geometric data with algebraic covers, yielding potential rigidity applications in higher Teichmüller-type settings.
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 $\varrho:π_1(X)\to G(K)$, we construct a $\varrho$-equivariant pluriharmonic map from the universal cover of $X$ into the Bruhat-Tits building $Δ(G)$ 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.