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Kodaira-Saito vanishing for the irregular Hodge filtration

Claude Sabbah

TL;DR

The paper extends Kodaira-Saito vanishing to the irregular Hodge filtration for irregular mixed Hodge modules by refining the foundational IrrMHM framework. It leverages Mochizuki21 to fix coherence issues via Malgrange extensions, rescalings along a tau-parameter, and tau-specializability, culminating in a robust category Cresc(X) and an improved IrrMHM(X). The main result demonstrates vanishings for the irregular Hodge graded de Rham complexes $\mathrm{gr}^{F^{irr}}\mathrm{pDR}(\mathcal{M})$ twisted by an ample line bundle, with symmetric vanishing for twists by $L^{-1}$ and Kollár-type pushforwards, and extends to log-geometry settings via $\Omega^p(\log D,\varphi,\alpha)$. A practical vanishing criterion (via a non-characteristic hyperplane and a cyclic cover) ensures the KS property for pushforwards, enabling broad geometric consequences. Together, these contributions provide a coherent irregular Hodge vanishing theory compatible with the standard functorial formalism of mixed twistor D-modules and sharpened by the Malgrange extension framework, significantly broadening the reach of Kodaira-Saito-type results in irregular Hodge theory.

Abstract

After making correct, and then improving, our definition of the category of irregular mixed Hodge modules thanks to Mochizuki's recent results arXiv:2108.03843, we show how these results allow us to obtain Kodaira-Saito-type vanishing theorems for the irregular Hodge filtration of irregular mixed Hodge modules.

Kodaira-Saito vanishing for the irregular Hodge filtration

TL;DR

The paper extends Kodaira-Saito vanishing to the irregular Hodge filtration for irregular mixed Hodge modules by refining the foundational IrrMHM framework. It leverages Mochizuki21 to fix coherence issues via Malgrange extensions, rescalings along a tau-parameter, and tau-specializability, culminating in a robust category Cresc(X) and an improved IrrMHM(X). The main result demonstrates vanishings for the irregular Hodge graded de Rham complexes twisted by an ample line bundle, with symmetric vanishing for twists by and Kollár-type pushforwards, and extends to log-geometry settings via . A practical vanishing criterion (via a non-characteristic hyperplane and a cyclic cover) ensures the KS property for pushforwards, enabling broad geometric consequences. Together, these contributions provide a coherent irregular Hodge vanishing theory compatible with the standard functorial formalism of mixed twistor D-modules and sharpened by the Malgrange extension framework, significantly broadening the reach of Kodaira-Saito-type results in irregular Hodge theory.

Abstract

After making correct, and then improving, our definition of the category of irregular mixed Hodge modules thanks to Mochizuki's recent results arXiv:2108.03843, we show how these results allow us to obtain Kodaira-Saito-type vanishing theorems for the irregular Hodge filtration of irregular mixed Hodge modules.
Paper Structure (18 sections, 11 theorems, 34 equations)

This paper contains 18 sections, 11 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Corrigenda to Bibi15
  3. Application of the results of Mochizuki21 to $\IrrMHM$
  4. An irregular Kodaira-Saito vanishing theorem
  5. Geometric consequences
  6. Notations and conventions
  7. A general framework for producing an irregular Hodge filtration
  8. A reminder on strict R-specializability
  9. Provisional corrected definition of IrrMHM
  10. Improved definition of IrrMHM
  11. Results of T. Mochizuki in Mochizuki21
  12. Specializability along tau0
  13. New definition of IrrMHM
  14. The category $\Cresc(X)$ and its properties
  15. Vanishing theorems
  16. ...and 3 more sections

Key Result

proposition 2

Assume that the $\wt R_X$-module $M$ is $R_X$-coherent. Then the $\wt\tauR_X(*\tau)$-module $\tauM$ is $\tauR'_X(*\tau)$-coherent, hence also $\wt\tauR_X(*\tau)$-coherent.

Theorems & Definitions (37)

  • proof
  • definition 1: Rescaling in the $\hb$-algebraic setting
  • proposition 2: Coherence
  • proof
  • proof
  • proposition 3
  • proof
  • definition 2
  • proposition 4
  • proof
  • ...and 27 more