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An iterative construction of complete Kähler--Einstein metrics

Quang-Tuan Dang, Tat Dat Tô

TL;DR

The paper develops a noncompact counterpart to Tsuji's iterative construction for complete Kähler–Einstein metrics with negative curvature under bounded geometry. By leveraging a uniform Bergman kernel expansion, it proves convergence of the iteration to a solution of $(dd^c phi_infty)^n = e^{phi_infty-phi_L}\u039f$ and obtains explicit rates, with faster convergence in the Kähler–Einstein case. It then shows that fiberwise KE metrics induce a positively curved metric on the relative canonical bundle $K_{X/Y}$ and that this positivity extends across singular fibers; the framework also handles cusp and mixed cone singularities and yields plurisubharmonic variation in families, including bounded pseudoconvex domains. Collectively, the results broaden KE metric variation theory to noncompact, bounded-geometry settings and provide tools for canonical metric constructions in fibred and degenerating settings.

Abstract

We extend Tsuji's iterative construction of complete Kähler--Einstein metrics with negative scalar curvature to noncompact Kähler manifolds with bounded geometry, using Berndtsson's method from the compact setting. Consequently, given a holomorphic surjective map $p:X\to Y$, where $X$ is a weakly pseudoconvex Kähler manifold and $Y$ is a complex manifold, and where the smooth fibers admit Kähler--Einstein metrics with negative scalar curvature and bounded geometry, we show that the fiberwise Kähler--Einstein metrics induce a semipositively curved metric on the relative canonical bundle $K_{X/Y}$. Moreover, our approach also applies to the plurisubharmonic variation of cusp Kähler--Einstein metrics.

An iterative construction of complete Kähler--Einstein metrics

TL;DR

The paper develops a noncompact counterpart to Tsuji's iterative construction for complete Kähler–Einstein metrics with negative curvature under bounded geometry. By leveraging a uniform Bergman kernel expansion, it proves convergence of the iteration to a solution of and obtains explicit rates, with faster convergence in the Kähler–Einstein case. It then shows that fiberwise KE metrics induce a positively curved metric on the relative canonical bundle and that this positivity extends across singular fibers; the framework also handles cusp and mixed cone singularities and yields plurisubharmonic variation in families, including bounded pseudoconvex domains. Collectively, the results broaden KE metric variation theory to noncompact, bounded-geometry settings and provide tools for canonical metric constructions in fibred and degenerating settings.

Abstract

We extend Tsuji's iterative construction of complete Kähler--Einstein metrics with negative scalar curvature to noncompact Kähler manifolds with bounded geometry, using Berndtsson's method from the compact setting. Consequently, given a holomorphic surjective map , where is a weakly pseudoconvex Kähler manifold and is a complex manifold, and where the smooth fibers admit Kähler--Einstein metrics with negative scalar curvature and bounded geometry, we show that the fiberwise Kähler--Einstein metrics induce a semipositively curved metric on the relative canonical bundle . Moreover, our approach also applies to the plurisubharmonic variation of cusp Kähler--Einstein metrics.

Paper Structure

This paper contains 12 sections, 18 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Notation and background
  3. General notation
  4. Bounded geometry on non-compact Kähler manifolds
  5. Uniform convergence of Tsuji’s iteration
  6. Asymptotic expansion of Bergman kernels
  7. Convergence of Tsuji’s iteration
  8. Variation of Kähler--Einstein metrics
  9. Positivity of relative Bergman kernels
  10. Plurisubharmonic variation of Kähler--Einstein metrics
  11. Families of Kähler--Einstein metrics on bounded pseudoconvex domains
  12. Families of Kähler--Einstein metrics with cusp singularities

Key Result

Theorem 1.1

Let $X$ be an $n$-dimensional Kähler manifold equipped with a positively curved line bundle $(L,e^{-\phi_L})$. Let $\Omega$ be a smooth volume form on $X$. Assume that there exists a smooth Hermitian metric $\phi_\infty$ on $L$ satisfying the complex Monge--Ampère equation Assume moreover that $(X,\omega_{\phi_\infty})$ is a Kähler manifold with bounded geometry of order $\ell\geq 5$. Let $\phi_1

Theorems & Definitions (32)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • ...and 22 more