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An $L^2$-$\partial\overline\partial$-Lemma on a class of complete Kähler manifolds

Riccardo Piovani

TL;DR

The paper addresses extending the classical $\partial\overline{\partial}$-Lemma, known for compact Kähler manifolds, to a non-compact, $L^2$-setting on complete Kähler manifolds by assuming a spectral gap for the Hodge Laplacian on $L^2$-forms. It develops a robust Hilbert-complex framework and proves spectral gaps for elliptic Aeppli and Bott-Chern Laplacians (via Kodaira-Spencer and Varouchas constructions), enabling closed-range results and elliptic-regularity arguments. The main result is an $L^2$-version of the $\partial\overline{\partial}$-Lemma: for $\alpha\in L^2A^k$ with $\partial\alpha=\overline{\partial}\alpha=0$, several equivalent representations exist, including $\alpha=\partial\overline{\partial}\beta$ for some $\beta\in L^2A^{k-2}$; the theorem generalizes the compact-case cohomological equivalences to a broad class of complete manifolds. This work provides a foundational tool for $L^2$-Hodge theory on complete Kähler spaces, with potential applications to $L^2$-cohomology, coverings, and domains with bounded geometry.

Abstract

We prove an $L^2$-$\partial\overline\partial$-Lemma involving smooth square integrable forms on complete Kähler manifolds, provided that the unique self-adjoint extension of the Hodge Laplacian on the Hilbert space of $L^2$-forms has a gap in its spectrum near zero. This generalises the classical $\partial\overline\partial$-Lemma on compact Kähler manifolds.

An $L^2$-$\partial\overline\partial$-Lemma on a class of complete Kähler manifolds

TL;DR

The paper addresses extending the classical -Lemma, known for compact Kähler manifolds, to a non-compact, -setting on complete Kähler manifolds by assuming a spectral gap for the Hodge Laplacian on -forms. It develops a robust Hilbert-complex framework and proves spectral gaps for elliptic Aeppli and Bott-Chern Laplacians (via Kodaira-Spencer and Varouchas constructions), enabling closed-range results and elliptic-regularity arguments. The main result is an -version of the -Lemma: for with , several equivalent representations exist, including for some ; the theorem generalizes the compact-case cohomological equivalences to a broad class of complete manifolds. This work provides a foundational tool for -Hodge theory on complete Kähler spaces, with potential applications to -cohomology, coverings, and domains with bounded geometry.

Abstract

We prove an --Lemma involving smooth square integrable forms on complete Kähler manifolds, provided that the unique self-adjoint extension of the Hodge Laplacian on the Hilbert space of -forms has a gap in its spectrum near zero. This generalises the classical -Lemma on compact Kähler manifolds.
Paper Structure (12 sections, 17 theorems, 66 equations)

This paper contains 12 sections, 17 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hilbert complexes
  4. Self-adjoint operators
  5. Closed image and spectral gap
  6. Strong and weak extensions
  7. From a differential complex to a Hilbert complex
  8. Differential complexes on complex manifolds
  9. Self-adjoint extensions of second- and fourth-order complex Laplacians
  10. Complete Kähler metrics
  11. Spectral gap of the elliptic Aeppli and Bott-Chern Laplacians
  12. L2-deldelbar-Lemma

Key Result

Theorem 1.1

Let $(M,g)$ be a complete Kähler manifold such that the self-adjoint Laplacian $\Delta$ has a spectral gap in $L^2\Lambda^{k}_{\mathbb C}$. Given a smooth $L^2$-form $\alpha\in L^2A^k_{\mathbb C}$ which satisfies $\partial\alpha={\overline{\partial}}\alpha=0$, then the following conditions are equiv

Theorems & Definitions (30)

  • Theorem 1.1: see Theorem \ref{['theorem l2 del delbar lemma']}
  • Lemma 2.1: BL
  • Theorem 2.2: RS2 or F
  • Theorem 2.3: F
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8: HP
  • ...and 20 more