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 ${\Pi }_g$ 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 Kähler manifold an irrational pencil, that is, a fibration onto a curve of genus $g\ge 2$, corresponds to a surjection of the profinite completion ${\hat{\pi }}_1(X)\twoheadrightarrow \widehat{{\Pi }_g}$, 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 Kähler manifold to be isomorphic to a product of curves of genera at least 2: if and only if ${\hat{\pi }}_1(X)\cong {\prod }_1^n\widehat{{\Pi }_{g_i}}$, 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.

9 pages, with an appendix by Pavel Zalesskii