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On the Hodge and V-filtrations of mixed Hodge modules

Dougal Davis, Ruijie Yang

TL;DR

The paper proves a Beilinson-type formula that expresses the Kashiwara–Malgrange $V$-filtration on complex mixed Hodge modules in terms of Hodge filtrations on localization, enabling a D-module lift to pro-mixed Hodge modules and yielding Beilinson-type descriptions of nearby cycles.It then connects this framework to birational geometry by showing higher multiplier ideals are recoverable as limits of Hodge ideals, establishing left-continuity as the precise obstruction to equality, and derives a birational transformation formula for higher multiplier ideals that extends the classical case.The authors provide streamlined vanishing results for Hodge ideals via twisted mixed Hodge modules and prove a functorial, derived pushforward formula for higher multiplier ideals under resolutions, showcasing the depth and flexibility of the mixed Hodge module approach in singularity theory.Together, these results deepen the relationship between Hodge-theoretic filtrations and multiplier theories, offering concrete computational tools and shedding light on foundational aspects of V-filtrations, nearby cycles, and birational invariants.

Abstract

In this paper, we prove a Beilinson-type formula for the V-filtration of Kashiwara and Malgrange on a complex mixed Hodge module, using Hodge filtrations on the localization. Our formula expresses the V-filtration as the filtered D-module underlying a pro-mixed Hodge module. We apply this to the theory of higher multiplier and Hodge ideals. Our first result shows that higher multiplier ideals can be obtained directly from Hodge ideals by taking a suitable limit. As a corollary, we deduce that Hodge ideals are left semi-continuous if and only if they coincide with higher multiplier ideals, thereby improving results of Saito and Mustaţă--Popa and resolving a folklore question. We further prove a birational transformation formula for higher multiplier ideals, generalising the classical formula for multiplier ideals and answering a question of Schnell and the second author. Finally, we provide very quick proofs of the main vanishing theorems for Hodge ideals, and strengthen a result of B. Chen.

On the Hodge and V-filtrations of mixed Hodge modules

TL;DR

The paper proves a Beilinson-type formula that expresses the Kashiwara–Malgrange $V$-filtration on complex mixed Hodge modules in terms of Hodge filtrations on localization, enabling a D-module lift to pro-mixed Hodge modules and yielding Beilinson-type descriptions of nearby cycles.It then connects this framework to birational geometry by showing higher multiplier ideals are recoverable as limits of Hodge ideals, establishing left-continuity as the precise obstruction to equality, and derives a birational transformation formula for higher multiplier ideals that extends the classical case.The authors provide streamlined vanishing results for Hodge ideals via twisted mixed Hodge modules and prove a functorial, derived pushforward formula for higher multiplier ideals under resolutions, showcasing the depth and flexibility of the mixed Hodge module approach in singularity theory.Together, these results deepen the relationship between Hodge-theoretic filtrations and multiplier theories, offering concrete computational tools and shedding light on foundational aspects of V-filtrations, nearby cycles, and birational invariants.

Abstract

In this paper, we prove a Beilinson-type formula for the V-filtration of Kashiwara and Malgrange on a complex mixed Hodge module, using Hodge filtrations on the localization. Our formula expresses the V-filtration as the filtered D-module underlying a pro-mixed Hodge module. We apply this to the theory of higher multiplier and Hodge ideals. Our first result shows that higher multiplier ideals can be obtained directly from Hodge ideals by taking a suitable limit. As a corollary, we deduce that Hodge ideals are left semi-continuous if and only if they coincide with higher multiplier ideals, thereby improving results of Saito and Mustaţă--Popa and resolving a folklore question. We further prove a birational transformation formula for higher multiplier ideals, generalising the classical formula for multiplier ideals and answering a question of Schnell and the second author. Finally, we provide very quick proofs of the main vanishing theorems for Hodge ideals, and strengthen a result of B. Chen.

Paper Structure

This paper contains 14 sections, 17 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Higher multiplier ideals as limits of Hodge ideals
  3. Vanishing theorems
  4. Functoriality of higher multiplier ideals
  5. Remarks on the foundations
  6. A Beilinson-type formula for the $V$-filtration
  7. The $V$-filtration along a smooth divisor
  8. Extension of the Hodge filtration
  9. The main theorem
  10. Applications to higher multiplier and Hodge ideals
  11. Higher multiplier and Hodge ideals
  12. Comparison between higher multiplier and Hodge ideals
  13. Vanishing theorems for Hodge ideals via twisted mixed Hodge modules
  14. Functoriality for higher multiplier ideals

Key Result

Theorem 1.1

Assume that $\mathcal{M}$ underlies a mixed Hodge module. Then for $\alpha \in {\mathbb{R}}$ and each $p\in \mathbb{Z}$, the inverse systems are constant for $n\gg 0$, where $F_\bullet$ denotes the Hodge filtration on the $!$ and $*$ extensions of the mixed Hodge modules $j^*{\mathcal{M}}[s]f^s/(s + \alpha)^n$. Moreover, eqn: Malgrange iso introduction induces isomorphisms of $\mathscr{D}_X[s]$-m

Theorems & Definitions (31)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6: Chenbingyi
  • Theorem 1.7
  • Proposition 1.8
  • Definition 2.1
  • Proposition 2.2
  • ...and 21 more