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Nash entropy, Calabi energy and geometric regularization of singular Kähler metrics

Bin Guo, Jian Song

TL;DR

This work connects analytic Sobolev bounds on singular Kähler spaces with synthetic Ricci curvature geometry by imposing uniform Nash entropy $\mathcal{N}_{\theta_X, p}(\omega) \le K$ and Calabi energy $\mathcal{Ca}(\omega) \le B$. It establishes uniform Laplacian and Green’s-function estimates, develops a regularization framework via resolutions that preserves these bounds, and uses twisted cscK metrics to approximate singular data. The main results show that the metric completion $(\hat X, d_\omega, \omega^n)$ is a non-collapsed ${\rm RCD}(-1,2n)$ space under a lower Ricci bound in the current sense, and is homeomorphic to the underlying projective variety $X$, with the regular set matching $X^\circ$ and controlled singular-set dimension. The paper also proves a Schwarz lemma and Lipschitz regularity for eigenfunctions in this singular setting, yielding abundant RCD examples topologically and holomorphically equivalent to projective varieties and linking complex-analytic regularization with metric-measure theory.

Abstract

We prove uniform Sobolev bounds for solutions of the Laplace equation on a general family of Kähler manifolds with bounded Nash entropy and Calabi energy. These estimates establish a connection to the theory of RCD spaces and provide abundant examples of RCD spaces topologically and holomorphically equivalent to projective varieties. Suppose $X$ is a normal projective variety that admits a resolution of singularities with relative nef or relative effective anti-canonical bundle. Then every admissible singular Kähler metric on $X$ with Ricci curvature bounded below induces a non-collapsed RCD space homeomorphic to the projective variety $X$ itself.

Nash entropy, Calabi energy and geometric regularization of singular Kähler metrics

TL;DR

This work connects analytic Sobolev bounds on singular Kähler spaces with synthetic Ricci curvature geometry by imposing uniform Nash entropy and Calabi energy . It establishes uniform Laplacian and Green’s-function estimates, develops a regularization framework via resolutions that preserves these bounds, and uses twisted cscK metrics to approximate singular data. The main results show that the metric completion is a non-collapsed space under a lower Ricci bound in the current sense, and is homeomorphic to the underlying projective variety , with the regular set matching and controlled singular-set dimension. The paper also proves a Schwarz lemma and Lipschitz regularity for eigenfunctions in this singular setting, yielding abundant RCD examples topologically and holomorphically equivalent to projective varieties and linking complex-analytic regularization with metric-measure theory.

Abstract

We prove uniform Sobolev bounds for solutions of the Laplace equation on a general family of Kähler manifolds with bounded Nash entropy and Calabi energy. These estimates establish a connection to the theory of RCD spaces and provide abundant examples of RCD spaces topologically and holomorphically equivalent to projective varieties. Suppose is a normal projective variety that admits a resolution of singularities with relative nef or relative effective anti-canonical bundle. Then every admissible singular Kähler metric on with Ricci curvature bounded below induces a non-collapsed RCD space homeomorphic to the projective variety itself.

Paper Structure

This paper contains 8 sections, 32 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Uniform linear estimates
  3. Regularization and the Lipchitz property for eigenfunctions
  4. Schwarz lemma
  5. Approximations by twisted cscK metrics
  6. Ricci curvature as a current
  7. Proof of Theorem \ref{['thm:main3']}: a special case
  8. Proof of Theorem \ref{['thm:main3']}: the general case

Key Result

Theorem 1.1

Let $(X, \theta_X)$ be an $n$-dimensional compact Kähler manifold equipped with a smooth Kähler metric $\theta_X$. Suppose $\omega$ is a Kähler metric in $\mathcal{V}(X, \theta_X, n, A, p, K)$ with $p>n$ satisfying and $u\in C^\infty(X)$ is a solution of the Laplace equation for $f\in C^\infty(X)$. Then the following hold.

Theorems & Definitions (61)

  • Theorem 1.1
  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • Remark 2.1
  • Proposition 2.2
  • ...and 51 more