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

TL;DR

This paper provides concise proofs of two theorems relating the profinite completion of the fundamental group to the geometric structure of compact Kähler manifolds, characterizing fibrations and products of curves.

Contribution

It introduces profinite versions of classical theorems, offering new criteria for identifying varieties isogenous to a product via fundamental group completions.

Findings

Profinite version of irrational pencil theorem with maximality property.

Characterization criterion for varieties isogenous to a product based on fundamental group.

Stronger results than recent existing theorems.

Abstract

We give a very short proof of two Theorems, whose content is outlined in the title, and where $\Pi_g$ 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 K\"ahler manifold an irrational pencil, that is, a fibration onto a curve of genus $g \geq 2$, corresponds to a surjection of the profinite completion $\widehat{\pi}_1(X) \twoheadrightarrow \widehat{\Pi_g}$, 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 K\"ahler manifold to be isomorphic to a product of curves of genera at least 2: if and only if $\widehat{\pi}_1(X) \cong \prod_1^n \widehat{\Pi_{g_i}}$, 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.