Paper · Nov 19, 2024

Dimension estimate and existence of holomorphic sections with polynomial growth on gradient Kähler Ricci shrinkers

Fei He, Jianyu Ou

math.DGarXiv:2411.12171v2
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Abstract

We prove an upper bound for the dimension of the linear space of holomorphic functions with polynomial growth on gradient Kähler Ricci shrinkers with bounded curvature. The upper bound is given as a power function of the growth rate. Similar results hold for holomorphic forms, and holomorphic sections of the pluri-anticanonical line bundle . We also prove the existence of holomorphic sections of with polynomial growth when the Kähler Ricci shrinker is asymptotically conical, provided is sufficiently large; as an application, we show that the Kodaira map constructed using such sections is a holomorphic embbedding into a complex projective space.

Comments: Minor corrections, added details, added reference and acknowledgement

Ricci solitonsKähler geometryBoundaries & non-compact

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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.

Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.