Rational Connectivity and Analytic Contractibility

TL;DR

This paper proves that for certain algebraic varieties over formal Laurent series, the Berkovich analytifications are homotopy equivalent if fibers are rationally connected, leading to invariance and contractibility results.

Contribution

It establishes homotopy equivalence of Berkovich analytifications under rational connectivity and shows contractibility for rationally connected varieties.

Findings

Homotopy equivalence of analytifications for rationally connected fibers

Invariance of homotopy type under birational transformations

Contractibility of analytifications of rationally connected varieties

Abstract

Let k be an algebraically closed field of characteristic 0, and let f be a morphism of smooth projective varieties from X to Y over the ring k((t)) of formal Laurent series. We prove that if a general geometric fiber of f is rationally connected, then the Berkovich analytifications of X and Y are homotopy equivalent. Two important consequences of this result are that the homotopy type of the Berkovich analytification of any smooth projective variety X over k((t)) is a birational invariant of X, and that the Berkovich analytification of a rationally connected smooth projective variety over k((t)) is contractible.