Kähler-Ricci shrinkers and Fano fibrations

Abstract

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

v2: minor revisions, with the addition of Proposition 3.10