Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

RCD structures on singular Kahler spaces of complex dimension three

Xin Fu, Bin Guo, Jian Song

Abstract

Let X be a projective variety of complex dimension 3 with log terminal singularities. We prove that every singular Kahler metric on X with bounded Nash entropy and Ricci curvature bounded below induces a compact RCD space homeomorphic to the projective variety X itself. In particular, singular Kahler-Einstein spaces of complex dimension 3 with bounded Nash entropy are compact RCD spaces topologically and holomorphically equivalent to the underlying projective variety. Various compactness theorems are also obtained for 3-dimensional projective varieties with bounded Ricci curvature. Such results establish connections among algebraic, geometric and analytic structures of klt singularities from birational geometry and provide abundant examples of RCD spaces from algebraic geometry via complex Monge-Ampere equations.

RCD structures on singular Kahler spaces of complex dimension three

Abstract

Let X be a projective variety of complex dimension 3 with log terminal singularities. We prove that every singular Kahler metric on X with bounded Nash entropy and Ricci curvature bounded below induces a compact RCD space homeomorphic to the projective variety X itself. In particular, singular Kahler-Einstein spaces of complex dimension 3 with bounded Nash entropy are compact RCD spaces topologically and holomorphically equivalent to the underlying projective variety. Various compactness theorems are also obtained for 3-dimensional projective varieties with bounded Ricci curvature. Such results establish connections among algebraic, geometric and analytic structures of klt singularities from birational geometry and provide abundant examples of RCD spaces from algebraic geometry via complex Monge-Ampere equations.

Paper Structure

This paper contains 19 sections, 70 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Ricci curvature for singular Kähler metrics
  3. Almost smooth Kähler spaces and RCD spaces
  4. Analytic $L^2$-estimates
  5. Proof of Theorem \ref{['mainthm1']}: a special case
  6. Approximating metrics on $Y$
  7. The RCD spaces $(Y_k, d_k, \mu_k)$
  8. The RCD space $(Y_\infty, d_\infty, \mu_\infty)$
  9. Proof of Theorem \ref{['mainthm1']}: the general case
  10. Singular Kähler metrics that are locally smoothable
  11. Dirichlet problems
  12. Proof of Theorem \ref{['locrcd7']}
  13. Proof of Theorem \ref{['mainthm2']}
  14. A generalization of Theorem \ref{['thm3spe']}
  15. A generalization of Theorem \ref{['thm3gene']}
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $X$ be an $n$-dimensional projective variety with log terminal singularities equipped with a smooth Kähler metric $\theta_X$. Suppose Then the metric measure space $(\hat{X}, d_\omega, \mu_\omega)$ induced by $(X, \omega)$ is a compact RCD space satisfying the following. Furthermore, if $\textnormal{Ric}(\omega)$ is also bounded above,

Theorems & Definitions (114)

  • Definition 1.1
  • Definition 1.2
  • Conjecture 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Definition 1.3
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.1
  • ...and 104 more