K\"ahler-Ricci shrinkers and Fano fibrations

TL;DR

This paper explores the relationship between K"ahler-Ricci shrinkers and algebraic geometry, establishing new conjectures linking their existence to algebraic stability conditions and analyzing singularity models.

Contribution

It proves that K"ahler-Ricci shrinkers are quasi-projective varieties and formulates conjectures connecting their existence and singularity structures to algebraic stability and degenerations.

Findings

K"ahler-Ricci shrinkers are quasi-projective varieties.

Conjecture linking K"ahler-Ricci shrinkers to K-stability of Fano fibrations.

Proposed connection between tangent flows at singularities and algebraic degenerations.

Abstract

In this paper, we build connections between K\"ahler-Ricci shrinkers, i.e., complete (possibly non-compact) shrinking gradient K\"ahler-Ricci solitons, and algebraic geometry. In particular, we (1). prove that a K\"ahler-Ricci shrinker is naturally a quasi-projective variety, using birational algebraic geometry; (2). formulate a conjecture relating the existence of K\"ahler-Ricci shrinkers and K-stability of polarized Fano fibrations, which unifies and extends the YTD type conjectures for K\"ahler-Einstein metrics, Ricci-flat K\"ahler cone metrics and compact K\"ahler-Ricci shrinkers; (3). formulate conjectures connecting tangent flows at singularities of K\"ahler-Ricci flows and algebraic geometry, via a 2-step degeneration for the weighted volume of a Fano fibration.