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$L^{2}$-approach to the Saito vanishing theorem

Hyunsuk Kim

TL;DR

The paper proves an analytic, $L^{2}$-based version of Saito’s vanishing theorem by linking it to a logarithmic de Rham framework and the classical Kodaira approach. It develops a complete analytic setup: bounded Higgs fields near SNC divisors, Deligne’s and prolongation extensions, and a carefully constructed complete Kähler metric on the open locus; these yield an $L^{2}$-Dolbeault resolution that computes the relevant hypercohomology and drives vanishing results. By proving a vanishing theorem for the graded logarithmic de Rham complex on a SNC-complement and leveraging Deligne extensions and prolongation, the authors reduce Saito’s vanishing to this analytic core and extend the argument to complex Hodge modules through standard $ ext{DR}$-functor and push-forward techniques. The approach not only recovers Saito’s vanishing but also provides a Kawamata–Viehweg-type perturbation, and it clarifies the interplay between $L^{2}$-techniques, Hodge theory, and $D$-module methods with potential generalizations to complex coefficients. This work thus offers a robust, analytic pathway to Saito vanishing that complements existing $ ext{D}$-module proofs and broadens the scope of positivity-based vanishing theorems in Hodge theory.

Abstract

We give an analytic proof of the Saito vanishing theorem using $L^{2}$-methods, by going back to the original idea for the proof of the Kodaira vanishing theorem.

$L^{2}$-approach to the Saito vanishing theorem

TL;DR

The paper proves an analytic, -based version of Saito’s vanishing theorem by linking it to a logarithmic de Rham framework and the classical Kodaira approach. It develops a complete analytic setup: bounded Higgs fields near SNC divisors, Deligne’s and prolongation extensions, and a carefully constructed complete Kähler metric on the open locus; these yield an -Dolbeault resolution that computes the relevant hypercohomology and drives vanishing results. By proving a vanishing theorem for the graded logarithmic de Rham complex on a SNC-complement and leveraging Deligne extensions and prolongation, the authors reduce Saito’s vanishing to this analytic core and extend the argument to complex Hodge modules through standard -functor and push-forward techniques. The approach not only recovers Saito’s vanishing but also provides a Kawamata–Viehweg-type perturbation, and it clarifies the interplay between -techniques, Hodge theory, and -module methods with potential generalizations to complex coefficients. This work thus offers a robust, analytic pathway to Saito vanishing that complements existing -module proofs and broadens the scope of positivity-based vanishing theorems in Hodge theory.

Abstract

We give an analytic proof of the Saito vanishing theorem using -methods, by going back to the original idea for the proof of the Kodaira vanishing theorem.
Paper Structure (21 sections, 27 theorems, 124 equations)

This paper contains 21 sections, 27 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Vanishing theorem for logarithmic de Rham complexes
  3. Comparison with previous results
  4. Preliminaries
  5. Complex Polarized Variation of Hodge Structures
  6. Bound for the Norm of Higgs field
  7. Prolongation via Norm Growth and the Nilpotent Orbit Theorem
  8. Some Standard Kähler Identities and Functional Analysis
  9. Proof of Theorem \ref{['theo-Main']}
  10. Dolbeault Resolution for the de Rham complex
  11. More Identities
  12. $L^{2}$-existence results
  13. Construction of the Kähler metric on the complement
  14. $L^{2}$-Dolbeault resolution
  15. Computation of $\mathcal{H}_{(r)}^{2} (E^{p,q} \otimes \mathcal{L}\otimes B)$ via Prolongation Bundles
  16. ...and 6 more sections

Key Result

Theorem 1.1

saito1990mixed*2.g Let $X$ be a complex projective variety and let $\mathcal{M} \in \mathop{\mathrm{MHM}}\nolimits(X)$ be a polarizable Hodge module on $X$. For an ample line bundle $\mathcal{L}$ on $X$, we have

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.3
  • Definition 2.4: Admissible Coordinates
  • Theorem 2.5: Mochizuki
  • Remark 2.6
  • ...and 24 more