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Profinite rigidity of Kähler groups: Riemann surfaces and subdirect products

Sam Hughes, Claudio Llosa Isenrich, Pierre Py, Matthew Stover, Stefano Vidussi

TL;DR

This work develops strong profinite rigidity phenomena for residually finite Kähler groups, showing that certain direct products of hyperbolic surface groups (and specific Kähler subgroups) are determined by their profinite completions within the class. The central tool is that holomorphic fibrations onto hyperbolic orbisurfaces, as well as the universal homomorphism to a direct product of such groups, are detectable at the level of profinite completions. It also proves profinite invariance of the BNS invariant in the Kähler setting, and derives rigidity and finiteness results for Kähler subgroups of direct products, including co-$\mathbb{Z}^2$ subgroups and torus-extended products. The results imply, for example, that certain aspherical smooth projective varieties are determined up to homeomorphism by their algebraic fundamental group, and they give a coherent framework for understanding when profinite data rigidly constrains Kähler and Kodaira-fibration groups, with implications for Grothendieck rigidity in this context.

Abstract

This paper establishes strong profinite rigidity results for Kähler groups, showing that certain groups are determined within the class of residually finite Kähler groups by their profinite completion. Examples include products of surface groups and certain groups with exotic finiteness properties studied earlier by Dimca-Papadima-Suciu and Llosa Isenrich. Consequently, there are aspherical smooth projective varieties that are determined up to homeomorphism by their algebraic fundamental group. The main tool is the following: the holomorphic fibrations of a closed Kähler manifold over hyperbolic 2-orbifolds can be recovered from the profinite completion of its fundamental group. We also prove profinite invariance of the BNS invariant.

Profinite rigidity of Kähler groups: Riemann surfaces and subdirect products

TL;DR

This work develops strong profinite rigidity phenomena for residually finite Kähler groups, showing that certain direct products of hyperbolic surface groups (and specific Kähler subgroups) are determined by their profinite completions within the class. The central tool is that holomorphic fibrations onto hyperbolic orbisurfaces, as well as the universal homomorphism to a direct product of such groups, are detectable at the level of profinite completions. It also proves profinite invariance of the BNS invariant in the Kähler setting, and derives rigidity and finiteness results for Kähler subgroups of direct products, including co- subgroups and torus-extended products. The results imply, for example, that certain aspherical smooth projective varieties are determined up to homeomorphism by their algebraic fundamental group, and they give a coherent framework for understanding when profinite data rigidly constrains Kähler and Kodaira-fibration groups, with implications for Grothendieck rigidity in this context.

Abstract

This paper establishes strong profinite rigidity results for Kähler groups, showing that certain groups are determined within the class of residually finite Kähler groups by their profinite completion. Examples include products of surface groups and certain groups with exotic finiteness properties studied earlier by Dimca-Papadima-Suciu and Llosa Isenrich. Consequently, there are aspherical smooth projective varieties that are determined up to homeomorphism by their algebraic fundamental group. The main tool is the following: the holomorphic fibrations of a closed Kähler manifold over hyperbolic 2-orbifolds can be recovered from the profinite completion of its fundamental group. We also prove profinite invariance of the BNS invariant.
Paper Structure (22 sections, 40 theorems, 82 equations)

This paper contains 22 sections, 40 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Orbisurfaces and ramified covers
  4. Direct products of groups and their subgroups
  5. The universal homomorphism
  6. Profinite rigidity of the universal homomorphism
  7. Preliminaries
  8. Homomorphisms to profinite orbisurface groups
  9. The universal homomorphism is a profinite invariant
  10. Profinite rigidity results for Kähler subgroups of direct products
  11. Preliminaries
  12. Virtual direct products
  13. Co-$\mathbb{Z}^2$ subgroups
  14. Once more with tori
  15. Central extensions and profinite groups
  16. ...and 7 more sections

Key Result

Theorem A

Let $X$ be a compact Kähler manifold with residually finite fundamental group. Assume that for some $r \ge 1$, where each $S_i$ is a closed oriented surface of genus at least $2$. Then $\pi_1(X)$ is isomorphic to $\prod_{i=1}^r \pi_1(S_i)$. Moreover, if $X$ is aspherical then it is biholomorphic to $\prod_{i=1}^r S_i$ for a suitable choice of complex structure on each of the $S_{i}$.

Theorems & Definitions (85)

  • Theorem A
  • Corollary 1.2
  • Definition 1.3
  • Theorem B
  • Definition 1.4
  • Remark 1.5
  • Definition 1.6
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • ...and 75 more