Notes on rational chain connectedness

TL;DR

This paper extends a key rational chain connectedness theorem to complex analytic spaces and shows that fibers of resolutions of certain singularities are rationally chain connected, using the minimal model program instead of extension theorems.

Contribution

It generalizes the rational chain connectedness theorem to the complex analytic setting and provides a new proof avoiding extension theorems, relying on the minimal model program.

Findings

Fibers of resolutions of complex analytic kawamata log terminal singularities are rationally chain connected.

The extension of the theorem broadens its applicability to complex analytic spaces.

A new proof approach using the minimal model program replaces the previous reliance on extension theorems.

Abstract

We extend Hacon--M\textsuperscript{c}Kernan's rational chain connectedness theorem to the complex analytic setting. As a consequence, we prove that the fibers of any resolution of singularities of complex analytic kawamata log terminal singularities are rationally chain connected. In contrast to the original approach, we avoid the use of extension theorems and instead rely on the minimal model program.