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K\"ahler-Ricci shrinkers and Fano fibrations
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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.
Forward citations
Cited by 2 Pith papers
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Uniqueness of shrinking K\"ahler-Ricci solitons on resolutions of K\"ahler cones
Every complete shrinking gradient Kähler-Ricci soliton on a resolution of a Kähler cone is asymptotically conical, which yields uniqueness up to pullback by biholomorphism.
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Compactness and Rigidity of Complete K\"ahler Ricci Shrinkers
A new first-order visibility argument derives compactness, splitting, and Gaussian rigidity for complete Kähler-Ricci shrinkers from the weight structure of their polarized Fano fibrations.
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