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$L^2$-Dolbeault resolutions and Nadel vanishing on weakly pseudoconvex complex spaces with singular Hermitian metrics

Yuta Watanabe

TL;DR

This work develops a general $L^2$-theory for the $\overline{\partial}$-operator on complex spaces with singular Hermitian metrics by constructing $L^2$-Dolbeault fine resolutions and isomorphisms and proving global $L^2$-estimates on the regular locus. The authors establish a robust framework involving canonical desingularizations, curvature currents, and multiplier ideal sheaves (with strong openness) to extend Nadel vanishing to weakly pseudoconvex spaces, including compact and non-compact settings, without requiring a global Kähler metric. Their approach yields vanishing results for $H^q(X,\omega_X^{GR}(h)\otimes\mathcal{O}(L))$ under broad hypotheses (e.g., singular positive metrics and big or nef/big line bundles) and provides a bridge between $L^2$-cohomology on the regular locus and global cohomology via the $L^2$-Dolbeault isomorphism. This advances vanishing theorems on singular spaces and broadens applicability to Moishezon, complete-intersection, and non-Kähler contexts with potential implications for complex-analytic and birational geometry.

Abstract

In this paper, in order to develop a more general $L^2$-theory for the $\overline{\partial}$-operator on complex spaces, we provide $L^2$-Dolbeault fine resolutions and isomorphisms, and $L^2$-estimates, for holomorphic line bundles on complex spaces equipped with singular Hermitian metrics. As applications, we obtain several generalizations of the Nadel vanishing theorem.

$L^2$-Dolbeault resolutions and Nadel vanishing on weakly pseudoconvex complex spaces with singular Hermitian metrics

TL;DR

This work develops a general -theory for the -operator on complex spaces with singular Hermitian metrics by constructing -Dolbeault fine resolutions and isomorphisms and proving global -estimates on the regular locus. The authors establish a robust framework involving canonical desingularizations, curvature currents, and multiplier ideal sheaves (with strong openness) to extend Nadel vanishing to weakly pseudoconvex spaces, including compact and non-compact settings, without requiring a global Kähler metric. Their approach yields vanishing results for under broad hypotheses (e.g., singular positive metrics and big or nef/big line bundles) and provides a bridge between -cohomology on the regular locus and global cohomology via the -Dolbeault isomorphism. This advances vanishing theorems on singular spaces and broadens applicability to Moishezon, complete-intersection, and non-Kähler contexts with potential implications for complex-analytic and birational geometry.

Abstract

In this paper, in order to develop a more general -theory for the -operator on complex spaces, we provide -Dolbeault fine resolutions and isomorphisms, and -estimates, for holomorphic line bundles on complex spaces equipped with singular Hermitian metrics. As applications, we obtain several generalizations of the Nadel vanishing theorem.
Paper Structure (13 sections, 28 theorems, 57 equations)

This paper contains 13 sections, 28 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Canonical desingularization of complex spaces
  4. Singular Hermitian metrics and its positivity on complex spaces
  5. Currents and its positivity on complex space
  6. Multiplier ideal sheaves and the strong openness property
  7. Positivity of curvature currents and bigness on compact space
  8. Exhaustion singular-positivity and Stein increasing on weakly pseudoconvex complex spaces
  9. Global $L^2$-estimates on the regular locus $X_{reg}$
  10. The weak $\overline{\partial}$-operator and its $L^2$-Dolbeault complex
  11. Nadel vanishing theorem on weakly pseudoconvex complex spaces
  12. Compact case
  13. Non-compact case

Key Result

Theorem 1.1

Let $X$ be a complex space of pure dimension $n$ and $L\longrightarrow X$ be a holomorphic line bundle equipped with a singular Hermitian metric $h$. If any local weight function of $h$ on $X$ is quasi-plurisubharmonic, then the $L^2$-Dolbeault complex is exact; that is, the complex $(\mathscr{L}^{n,\ast}_{L,h},\overline{\partial})$ is $L^2$-Dolbeault fine resolution of $\omega_X^{GR}(h)\otimes\m

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 2.1: BM97, cf. Hir64
  • Lemma 2.2
  • Definition 2.3: GPR94, Fuj75
  • ...and 46 more