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

Fei He, Jianyu Ou

TL;DR

The paper analyzes holomorphic objects on gradient Kähler Ricci shrinkers with bounded curvature, establishing finite-dimensional bounds for spaces of holomorphic sections with polynomial growth via spectral counts of drifted elliptic operators. It extends these bounds to holomorphic $(p,0)$-forms and to sections of $K_M^{-q}$, and obtains precise growth-dependent dimension estimates. For asymptotically conical shrinkers, the authors prove the existence of nontrivial holomorphic sections of $K_M^{-q}$ with polynomial growth for large $q$, using $L^2$ methods and pullbacks along the $-\nabla f$ flow, and they show the Kodaira map yields a holomorphic embedding into projective space. Together, these results connect analytic techniques (spectral theory, frequency functions, $f$-Laplacians) with complex-geometric consequences (separation, embedding) under curvature and asymptotic decay hypotheses.

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 $(p, 0)-$forms, and holomorphic sections of the pluri-anticanonical line bundle $K_M^{-q}$. We also prove the existence of holomorphic sections of $K_M^{-q}$ with polynomial growth when the Kähler Ricci shrinker is asymptotically conical, provided $q$ 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.

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

TL;DR

The paper analyzes holomorphic objects on gradient Kähler Ricci shrinkers with bounded curvature, establishing finite-dimensional bounds for spaces of holomorphic sections with polynomial growth via spectral counts of drifted elliptic operators. It extends these bounds to holomorphic -forms and to sections of , and obtains precise growth-dependent dimension estimates. For asymptotically conical shrinkers, the authors prove the existence of nontrivial holomorphic sections of with polynomial growth for large , using methods and pullbacks along the flow, and they show the Kodaira map yields a holomorphic embedding into projective space. Together, these results connect analytic techniques (spectral theory, frequency functions, -Laplacians) with complex-geometric consequences (separation, embedding) under curvature and asymptotic decay hypotheses.

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.

Paper Structure

This paper contains 12 sections, 19 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. The spectral counting number
  3. Definition and an upper estimate for the frequency.
  4. Estimate of the spectral counting number
  5. Holomorphic sections
  6. General discussion
  7. Holomorphic $(p, 0)$-forms.
  8. Holomorphic sections of $K_M^{-q}$.
  9. Frequency on asymptotically conical gradient Kähler Ricci shrinker
  10. Definitions and basic properties
  11. Asymptotically conical shrinkers
  12. Existence of holomorphic sections with polynomial growth

Key Result

Theorem 1.1

Suppose $(M^n, g, f)$ is a complete gradient shrinking Ricci soliton with bounded curvature, $E \to M$ is a vector bundle equipped with a metric $\langle, \rangle$ and a compatible connection. Let $L = - \Delta + \nabla_{\nabla f}+ A$, where $\Delta$ is the rough Laplacian and $A\in \Gamma(End(E))$ where $C$ is a constant depending on the geometry.

Theorems & Definitions (37)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 2
  • Remark 3
  • Theorem 2.1
  • proof
  • ...and 27 more