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Weight filtration and generating level

Henry Dakin

TL;DR

This work studies the canonical complex mixed Hodge module structure on the left $\\mathscr{D}_X$-module $\\mathscr{M}(f^{-\\alpha})$ and analyzes its Hodge filtration $F^H_\\bullet$ and weight filtration $W_\\bullet$. It develops two complementary frameworks: (i) log-resolution methods to express filtrations in terms of pushforwards and the Kashiwara-Malgrange $V$-filtration, and (ii) algebraic and microlocal approaches via the $V$-filtration, nearby/vanishing cycles, and microlocal invariants such as weighted minimal exponents. A central result is a general formula linking the Hodge and weight filtrations to the $V$-filtration data: $F_k^H W_{n+l}\\mathscr{M}(f^{-\\alpha}) =\\psi_0( F_k^{t-ord}K_lV^0 i_{f,+}\\mathscr{M}(f^{-\\alpha}) )$, from which corollaries connect to multiplier/adjoint ideals and birational thresholds. The paper then introduces weighted minimal exponents and weighted microlocal multiplier ideals to bound the generating level and the highest weight of filtration steps, with explicit computations for SNC and weighted homogeneous singularities and applications to parametrically prime divisors; together, these results deepen the link between Hodge theory, D-module theory, and birational geometry. The techniques yield practical criteria and explicit formulas for the filtrations in broad classes of divisors, highlighting the interplay between geometric singularities and Hodge-theoretic invariants.

Abstract

We study the canonical mixed Hodge module structure associated to the $\mathscr{D}_X$-module $\mathscr{M}(f^{-α}):=\mathscr{O}_X(*f)f^{-α}$. We particularly focus on the weight filtration and extend many known results to the weighted setting. We obtain new relations between Hodge theory and birational geometry. We derive a general formula for the Hodge and weight filtrations on $\mathscr{M}(f^{-α})$, and use this to obtain results concerning the largest weight of $\mathscr{M}(f^{-α})$ and the generating level of weight filtration steps. Finally, we obtain expressions for several classes of divisor, including certain parametrically prime divisors.

Weight filtration and generating level

TL;DR

This work studies the canonical complex mixed Hodge module structure on the left -module and analyzes its Hodge filtration and weight filtration . It develops two complementary frameworks: (i) log-resolution methods to express filtrations in terms of pushforwards and the Kashiwara-Malgrange -filtration, and (ii) algebraic and microlocal approaches via the -filtration, nearby/vanishing cycles, and microlocal invariants such as weighted minimal exponents. A central result is a general formula linking the Hodge and weight filtrations to the -filtration data: , from which corollaries connect to multiplier/adjoint ideals and birational thresholds. The paper then introduces weighted minimal exponents and weighted microlocal multiplier ideals to bound the generating level and the highest weight of filtration steps, with explicit computations for SNC and weighted homogeneous singularities and applications to parametrically prime divisors; together, these results deepen the link between Hodge theory, D-module theory, and birational geometry. The techniques yield practical criteria and explicit formulas for the filtrations in broad classes of divisors, highlighting the interplay between geometric singularities and Hodge-theoretic invariants.

Abstract

We study the canonical mixed Hodge module structure associated to the -module . We particularly focus on the weight filtration and extend many known results to the weighted setting. We obtain new relations between Hodge theory and birational geometry. We derive a general formula for the Hodge and weight filtrations on , and use this to obtain results concerning the largest weight of and the generating level of weight filtration steps. Finally, we obtain expressions for several classes of divisor, including certain parametrically prime divisors.

Paper Structure

This paper contains 13 sections, 67 theorems, 323 equations.

Table of Contents

  1. Introduction
  2. Hodge filtrations via log resolutions
  3. Hodge module prerequisites
  4. The complex mixed Hodge module structure on $\mathscr{M}(f^{-\alpha})$
  5. Hodge and weight filtrations via log resolutions
  6. Hodge filtrations via canonical $V$-filtrations
  7. Specialisability and quasi-unipotency
  8. Hodge and weight filtrations via $V$-filtrations
  9. Examples: SNC and weighted homogeneous isolated singularities
  10. Microlocalisation and weighted minimal exponents
  11. Weighted minimal exponents and microlocal multiplier ideals
  12. Generating level and highest weights for $M(f^{-\alpha})$
  13. Parametrically prime divisors

Key Result

Theorem 1.1

Write $\mathscr{M}_r(f^{-\alpha})$ for the right $\mathscr{D}_X$-module associated to $\mathscr{M}(f^{-\alpha})$, with induced Hodge and weight filtrations. Then

Theorems & Definitions (159)

  • Theorem 1.1: Proposition \ref{['propSNCresW']} and Theorem \ref{['thmpiplus']} below
  • Corollary 1.2: Corollary \ref{['corgenlevWn']} below
  • Definition 1.3
  • Corollary 1.4: Corollary \ref{['corpiplus']} below
  • Corollary 1.5: Corollary \ref{['corpair']} below
  • Theorem 1.6: Theorem \ref{['thmmainformula']} below
  • Definition 1.7
  • Proposition 1.8: Proposition \ref{['prophodgepole2']} below
  • Corollary 1.9: Corollary \ref{['corlct']} below
  • Theorem 1.10: Theorem \ref{['thmlargestweight']} below
  • ...and 149 more