When is the diagonal contractible?

TL;DR

This paper characterizes when the diagonal of a smooth projective complex variety can be contracted via a birational morphism, linking it to the variety's Albanese dimension and irregularity.

Contribution

It provides necessary and sufficient conditions for contracting the diagonal of a variety through birational morphisms, extending understanding of diagonal contractions.

Findings

Existence of birational morphism contracting the diagonal is equivalent to maximal Albanese dimension and high irregularity.

Conditions for arbitrary diagonal contractions are established.

Criteria for birational morphisms that are isomorphisms outside the diagonal are given.

Abstract

For a smooth projective complex variety $X$, we prove that there exists a birational morphism $X\times X\to Y$ to a projective variety $Y$ contracting the diagonal $\Delta_X\subset X\times X$ to a point if and only if $X$ has maximal Albanese dimension and irregularity $q\geq 2\dim X$. We also give necessary and sufficient conditions for arbitrary contractions of the diagonal and for the existence of a birational morphism which is an isomorphism outside the diagonal.