2024/10/12 by Song Sun, Sun, Song, Junsheng Zhang +1 · 4 citations
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2410.09661
openalex publication_date 2024/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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.