Kähler-Ricci shrinkers and Fano fibrations
Song Sun, Junsheng Zhang
TL;DR
This work establishes a foundational bridge between Kähler-Ricci shrinkers and algebraic geometry by proving that shrinkers define quasi-projective varieties and polarized Fano fibrations, using Birkar’s boundedness rather than analytical compactifications. It introduces a unifying weighted-volume functional $\mathbb{W}$ for Fano fibrations, showing how it subsumes normalized volumes and $\tilde{\beta}$-invariants, and proves a convexity principle that yields a unique minimizer linked to K-stability. The authors formulate a comprehensive YTD-type conjecture for the existence and uniqueness of shrinkers on polarized Fano fibrations, along with a two-step degeneration mechanism that connects local tangent flows of Kähler-Ricci flow to algebro-geometric degenerations. They further develop a program for moduli, finite-time singularities, and tangent flows at infinity within this framework, aiming to unify several canonical metric theories (Kähler–Einstein, Ricci-flat cones, shrinkers) under a single algebro-geometric stability paradigm. If realized, this program would provide a powerful, unifying lens for the long-standing links between geometric analysis and birational geometry in both compact and non-compact settings.
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.