As on arXiv
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.
Comments: v2: minor revisions, with the addition of Proposition 3.10
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
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Singular Kähler–Ricci shrinkers and polarized Fano fibrations
Topics Kähler–Ricci flow, Kähler geometry, Ricci solitons · Same author Junsheng Zhang · Shared terms “Fano fibration”, “Ricci shrinkers”, “polarized Fano”
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Scalar curvature bounds and tangent cones for Kähler–Ricci shrinkers
Topics Boundaries & non-compact, Kähler geometry, Ricci solitons · Same author Junsheng Zhang · Shared terms “Ricci shrinkers”, “Fano fibration”, “polarized Fano”
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Gromov-Hausdorff Limits of Noncollapsed Kähler-Ricci Flows and the Geometry of Ricci Shrinkers
Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Junsheng Zhang · Shared terms “Ricci shrinkers”, “Fano fibration”, “Kähler cone”
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A Matsushima theorem, uniqueness and K-polystability for Kähler-Ricci shrinkers
Topics Boundaries & non-compact, Kähler geometry, Stability & uniqueness · Same author Junsheng Zhang · Shared terms “Ricci shrinkers”, “Fano fibration”, “polarized Fano”
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Weak transcendental base-point freeness and diameter lower bounds for the Kähler-Ricci flow
Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Junsheng Zhang · Shared terms “Fano”, “compact Kähler”
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Uniqueness of asymptotically conical shrinking gradient Kähler-Ricci solitons
Topics Ricci-flat & Calabi–Yau, Kähler–Ricci solitons, Boundaries & non-compact · Shared terms “Fano fibration”, “shrinking gradient”, “gradient Kähler”
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Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I
Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Junsheng Zhang · Shared terms “compact Kähler”, “tangent flow”
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K-polystability of Asymptotically Conical Kähler-Ricci Shrinkers
Topics Boundaries & non-compact, Kähler geometry, Singularities · Shared terms “Ricci shrinkers”, “Fano fibration”, “polarized Fano”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 5A.6Where This Track Leads
The Kähler–Ricci flow, and when to come back to it.
- 11B.4Blow-Ups, Blow-Downs and Splitting at Infinity
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.
- 11B.6Rotationally Symmetric Flows and the Standard Solution
The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.