K\"ahler-Ricci shrinkers and Fano fibrations

Song Sun; Junsheng Zhang

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arXiv · 2410.09661

K\"ahler-Ricci shrinkers and Fano fibrations

Song Sun·Junsheng Zhang

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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.

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BibTeX RIS

Song Sun, Junsheng Zhang. 2024-10-12. K\"ahler-Ricci shrinkers and Fano fibrations. https://arxiv.org/abs/2410.09661

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