Pluriharmonic maps into buildings and symmetric differentials

TL;DR

This paper constructs equivariant harmonic maps into Bruhat-Tits buildings for quasi-projective varieties and uses this to establish the existence of nonzero symmetric differentials under certain conditions, extending previous results.

Contribution

It generalizes the construction of harmonic maps into buildings and the existence of symmetric differentials to the quasi-projective case, broadening the scope of prior work.

Findings

Constructed $ ho$-equivariant harmonic maps into Bruhat-Tits buildings for quasi-projective varieties.

Proved the existence of nonzero global logarithmic symmetric differentials under linear representations with infinite image.

Extended Gromov-Schoen and Brunebarbe-Klingler-Totaro results to the quasi-projective setting.

Abstract

Given a complex smooth quasi-projective variety $X$, a semisimple 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 (pluri-)harmonic map from the universal cover of $X$ into the Bruhat-Tits building $\Delta(G)$ of $G$, with some suitable asymptotic behavior. This theorem generalizes the previous work by Gromov-Schoen to the quasi-projective setting. As an application, we prove that $X$ has nonzero global logarithmic symmetric differentials if there exists a linear representation $\pi_1(X)\to {\rm GL}_N(\mathbb{K})$ with infinite image, where $ \mathbb{K}$ is any field. This theorem generalizes the previous work by Brunebarbe, Klingler and Totaro to the quasi-projective setting.