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

Abstract: We give a very short proof of two Theorems, whose content is outlined in the title, and where $Πg$ 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.