Fundamental group and pluridifferentials on compact K\"ahler manifolds

TL;DR

This paper establishes conditions under which compact Kähler manifolds are simply-connected based on their symmetric cotangent algebra, and explores implications for rational connectedness and fundamental group isomorphisms in holomorphic maps.

Contribution

It proves that a compact Kähler manifold with trivial symmetric cotangent algebra is simply-connected and extends this to relative cases involving holomorphic maps and fundamental groups.

Findings

Compact Kähler manifolds with trivial symmetric cotangent algebra are simply-connected.

Proper surjective holomorphic maps with certain fiber conditions induce fundamental group isomorphisms.

Conjecture that such manifolds are rationally connected.

Abstract

A compact K\"ahler manifold is shown to be simply-connected if its `symmetric cotangent algebra' is trivial. Conjecturally, such a manifold should even be rationally connected. The relative version is also shown: a proper surjective connected holomorphic map between connected complex manifolds induces an isomorphism of fundamental groups if its smooth fibres are as above, and if the domain is K\"ahler.