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Irrational pencils, and characterization of Varieties isogenous to a product, via the Profinite completion of the Fundamental group

Fabrizio Catanese, appendix by Pavel Zalesskii

TL;DR

The paper provides concise proofs of profinite analogs for two central results on complex geometry: (i) irrational pencils correspond to maximal surjections from the profinite fundamental group to a profinite surface group, and (ii) a compact Kähler manifold is isomorphic to a product of curves of genus at least 2 precisely when its profinite fundamental group is a product of the corresponding profinite surface groups, subject to volume/cohomology criteria. The approach translates geometric fibrations into profinite group data, leveraging the profinite completion, residual finiteness, and an appendix result that rules out certain splitting maps. The main contribution is a streamlined, profinite-proof framework that strengthens existing theorems and clarifies the role of the algebraic fundamental group in characterizing varieties isogenous to a product. These results have implications for understanding the structure of VIPs through group-theoretic invariants, and for Bridging complex geometry with profinite methods.

Abstract

We give a very short proof of two Theorems, whose content is outlined in the title, and where $Π_g$ is the fundamental group of a compact complex curve of genus $g$: (1) Theorem 2.1 of the irrational pencil in the profinite version, saying that for a compact Kähler manifold an irrational pencil, that is, a fibration onto a curve of genus $g \geq 2$, corresponds to a surjection of the profinite completion $\widehatπ_1(X) \twoheadrightarrow \widehat{Π_g}$, which satisfies a maximality property; (2) Theorem 1.4 on the characterization of varieties isogenous to a product, profinite version, giving in particular a criterion for $X$ a compact Kähler manifold to be isomorphic to a product of curves of genera at least 2: if and only if $\widehatπ_1(X) \cong \prod_1^n \widehat{Π_{g_i}}$, and some volume or cohomological condition is satisfied. Theorem 1.4 yields a stronger result than the Main Theorem A of a recent article by 5 authors.

Irrational pencils, and characterization of Varieties isogenous to a product, via the Profinite completion of the Fundamental group

TL;DR

The paper provides concise proofs of profinite analogs for two central results on complex geometry: (i) irrational pencils correspond to maximal surjections from the profinite fundamental group to a profinite surface group, and (ii) a compact Kähler manifold is isomorphic to a product of curves of genus at least 2 precisely when its profinite fundamental group is a product of the corresponding profinite surface groups, subject to volume/cohomology criteria. The approach translates geometric fibrations into profinite group data, leveraging the profinite completion, residual finiteness, and an appendix result that rules out certain splitting maps. The main contribution is a streamlined, profinite-proof framework that strengthens existing theorems and clarifies the role of the algebraic fundamental group in characterizing varieties isogenous to a product. These results have implications for understanding the structure of VIPs through group-theoretic invariants, and for Bridging complex geometry with profinite methods.

Abstract

We give a very short proof of two Theorems, whose content is outlined in the title, and where is the fundamental group of a compact complex curve of genus : (1) Theorem 2.1 of the irrational pencil in the profinite version, saying that for a compact Kähler manifold an irrational pencil, that is, a fibration onto a curve of genus , corresponds to a surjection of the profinite completion , which satisfies a maximality property; (2) Theorem 1.4 on the characterization of varieties isogenous to a product, profinite version, giving in particular a criterion for a compact Kähler manifold to be isomorphic to a product of curves of genera at least 2: if and only if , and some volume or cohomological condition is satisfied. Theorem 1.4 yields a stronger result than the Main Theorem A of a recent article by 5 authors.

Paper Structure

This paper contains 6 sections, 10 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Irrational pencils Theorem via the algebraic fundamental group
  3. Profinite completion
  4. Proof of Theorem \ref{['PIP']}
  5. Proof of Theorem \ref{['PVIP']}
  6. Appendix by Pavel Zalesskii

Key Result

Theorem 1.2

Let $X$ be a compact Kähler manifold of dimension $n$ such that (i) $\pi_1(X) \cong \Gamma$ admits an index $d$ subgroup $\Gamma'$ such that where $\Pi_g$ is the fundamental group of a compact complex curve of genus $g$, and all $g_j \geq 2$, (ii) $H^{2n}(X, \mathbb{Z})$ has index $d$ inside the image of $H^{2n}(\Gamma', \mathbb{Z})$ in $H^{2n}(X', \mathbb{Z})$, $X' \rightarrow X$ being the unram

Theorems & Definitions (22)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 12 more