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Pluriharmonic maps into buildings and symmetric differentials

Damian Brotbek, Georgios Daskalopoulos, Ya Deng, Chikako Mese

TL;DR

This work extends Gromov–Schoen's non-archimedean harmonic-map theory to smooth complex quasi-projective varieties by constructing a $\varrho$-equivariant pluriharmonic map $\tilde{u}:\widetilde{X}\to\Delta(G)$ into the Bruhat–Tits building $\Delta(G)$ for a Zariski-dense $\varrho:\pi_1(X)\to G(K)$, with local Lipschitz regularity and logarithmic energy growth. It proves finiteness of energy near the boundary under quasi-unipotent monodromy and shows functoriality under pullback along morphisms, thereby enabling a robust quasi-projective analogue of the Gromov–Schoen machinery. As an application, the authors show that if there exists a linear representation $\pi_1(X)\to\mathrm{GL}_N(\mathbb{K})$ with infinite image, then $H^0(\overline{X},\mathrm{Sym}^k\Omega_{\overline{X}}(\log\Sigma))\neq0$ for some $k>0$, extending Brunebarbe–Klingler–Totaro to quasi-projective settings. The results bridge non-archimedean harmonic maps with geometric-differential consequences, and connect rigidity/integrality phenomena (Simpson's conjecture) to the existence of logarithmic symmetric differentials, yielding significant implications for the structure of fundamental groups of quasi-projective varieties.

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 $\varrho:π_1(X)\to G(K)$, we construct a $\varrho$-equivariant (pluri-)harmonic map from the universal cover of $X$ into the Bruhat-Tits building $Δ(G)$ 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 $π_1(X)\to {\rm GL}_N(\mathbb{K})$ 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.

Pluriharmonic maps into buildings and symmetric differentials

TL;DR

This work extends Gromov–Schoen's non-archimedean harmonic-map theory to smooth complex quasi-projective varieties by constructing a -equivariant pluriharmonic map into the Bruhat–Tits building for a Zariski-dense , with local Lipschitz regularity and logarithmic energy growth. It proves finiteness of energy near the boundary under quasi-unipotent monodromy and shows functoriality under pullback along morphisms, thereby enabling a robust quasi-projective analogue of the Gromov–Schoen machinery. As an application, the authors show that if there exists a linear representation with infinite image, then for some , extending Brunebarbe–Klingler–Totaro to quasi-projective settings. The results bridge non-archimedean harmonic maps with geometric-differential consequences, and connect rigidity/integrality phenomena (Simpson's conjecture) to the existence of logarithmic symmetric differentials, yielding significant implications for the structure of fundamental groups of quasi-projective varieties.

Abstract

Given a complex smooth quasi-projective variety , a semisimple algebraic group defined over some non-archimedean local field and a Zariski dense representation , we construct a -equivariant (pluri-)harmonic map from the universal cover of into the Bruhat-Tits building of , 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 has nonzero global logarithmic symmetric differentials if there exists a linear representation with infinite image, where is any field. This theorem generalizes the previous work by Brunebarbe, Klingler and Totaro to the quasi-projective setting.
Paper Structure (38 sections, 37 theorems, 290 equations)

This paper contains 38 sections, 37 theorems, 290 equations.

Table of Contents

  1. Introduction
  2. Main theorem
  3. An application
  4. Notation and Convention
  5. Preliminaries
  6. NPC spaces and Euclidean buildings
  7. Harmonic maps to NPC spaces
  8. Admissible coordinates
  9. Existence of Harmonic maps to Bruhat-Tits buildings
  10. Preliminary lemmas
  11. Equivariant maps and sections
  12. Pullback bundles
  13. Regularity results of Gromov-Schoen
  14. A Bertini-type theorem
  15. Logarithmic energy growth (I)
  16. ...and 23 more sections

Key Result

Theorem 1

Let $X$ be a complex smooth quasi-projective variety, and let $G$ be a semisimple algebraic group defined over a non-archimedean local field $K$. Denote by $\widetilde{X}$ the universal cover of $X$. If $\varrho: \pi_1(X) \to G(K)$ is a Zariski-dense representation, then there exists a $\varrho$-equ

Theorems & Definitions (103)

  • Theorem 1: =\ref{['thm:harmonicmaps', 'logestimate', 'thm:functorial']}
  • Theorem 1: BKT13
  • Theorem 2
  • Definition 1.1: Geodesic space
  • Definition 1.2: NPC space
  • Definition 1.3: Harmonic maps
  • Definition 1.4: Locally Lipschitz
  • Remark 1.5
  • Definition 1.6
  • Theorem 2.1: Existence of (pluri-)harmonic maps
  • ...and 93 more