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arxiv: 2309.16763 · v7 · submitted 2023-09-28 · 🧮 math.AG · math.AC· math.CV

Higher multiplier ideals

Christian Schnell , Ruijie Yang This is my paper

Pith reviewed 2026-05-24 06:16 UTC · model grok-4.3

classification 🧮 math.AG math.ACmath.CV
keywords higher multiplier idealsmixed Hodge modulesV-filtrationmultiplier idealstheta divisorsabelian varietiesvanishing theoremsrestriction theorems
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The pith

Higher multiplier ideals form a two-parameter family of ideal sheaves for any Q-effective divisor on a complex manifold via mixed Hodge modules.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper associates to any Q-effective divisor on a complex manifold a family of ideal sheaves called higher multiplier ideals. These ideals are indexed by an integer for the Hodge level and a rational number, and they carry a weight filtration. When the Hodge level is zero the new ideals coincide with the classical multiplier ideals. The authors prove vanishing theorems and restriction theorems for the family, give criteria for nontriviality, and define the center of minimal exponent as a generalization of the minimal log canonical center. The construction rests on twisted Hodge modules to access the global structure of the V-filtration, and it yields new results on singularities of theta divisors.

Core claim

By exploiting the global structure of the V-filtration along an effective divisor with the notion of twisted Hodge modules, the authors associate to any Q-effective divisor a family of ideal sheaves called higher multiplier ideals. This family is indexed by an integer Hodge level and a rational number, admits a weight filtration, and recovers the usual multiplier ideals when the Hodge level is zero. The construction allows systematic study of local and global properties including vanishing theorems, restriction theorems, and criteria for nontriviality, and generalizes the minimal log canonical center to the center of minimal exponent.

What carries the argument

Higher multiplier ideals, a family of ideal sheaves indexed by Hodge level and rational parameter and constructed via mixed Hodge modules and V-filtrations on twisted Hodge modules.

If this is right

  • Vanishing theorems hold for the higher multiplier ideals at each Hodge level.
  • Restriction theorems apply to the higher multiplier ideals under suitable morphisms.
  • Criteria determine when the higher multiplier ideals are nontrivial.
  • The center of minimal exponent generalizes the notion of minimal log canonical center.
  • New cases of conjectures on singularities of theta divisors on principally polarized abelian varieties are proved.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The weight filtration on higher multiplier ideals may supply additional numerical invariants for measuring singularities.
  • The twisted Hodge module technique could adapt to other divisor-related filtrations in algebraic geometry.
  • The nontriviality criteria might help classify divisors with prescribed singularity types.
  • Applications could extend beyond abelian varieties to singularities arising in other families of varieties.

Load-bearing premise

The standard properties of mixed Hodge modules and V-filtrations extend to twisted Hodge modules in a way that produces ideal sheaves satisfying the vanishing, restriction, and nontriviality properties.

What would settle it

An explicit computation on a concrete Q-effective divisor where a higher multiplier ideal at positive Hodge level fails a stated vanishing theorem or restriction theorem would show the construction does not hold.

read the original abstract

We associate a family of ideal sheaves to any Q-effective divisor on a complex manifold, called higher multiplier ideals, using the theory of mixed Hodge modules and V-filtrations. This family is indexed by two parameters, an integer indicating the Hodge level and a rational number, and these ideals admit a weight filtration. When the Hodge level is zero, they recover the usual multiplier ideals. We study the local and global properties of higher multiplier ideals systematically. In particular, we prove vanishing theorems and restriction theorems, provide criteria for the nontriviality, and introduce the center of minimal exponent (generalizing the notion of minimal log canonical center). The main idea is to exploit the global structure of the V-filtration along an effective divisor using the notion of twisted Hodge modules. As applications, we prove new cases of conjectures by Debarre, Casalaina-Martin and Grushevsky on singularities of theta divisors on principally polarized abelian varieties.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes a new family of ideal sheaves called higher multiplier ideals for Q-effective divisors on complex manifolds, constructed via mixed Hodge modules and V-filtrations. These ideals are parametrized by an integer (Hodge level) and a rational number, admit a weight filtration, and reduce to classical multiplier ideals when the Hodge level is zero. The paper claims to establish vanishing and restriction theorems, nontriviality criteria, and the notion of the center of minimal exponent, with applications to conjectures on singularities of theta divisors.

Significance. If the stated properties hold, the construction would provide a two-parameter generalization of multiplier ideals that incorporates Hodge-theoretic data, potentially yielding refined vanishing and restriction results together with a new notion of minimal center. The applications to the Debarre–Casalaina-Martin–Grushevsky conjectures indicate possible utility in the study of theta divisors on principally polarized abelian varieties.

major comments (1)
  1. [Abstract] Abstract: the central claims (vanishing theorems, restriction theorems, nontriviality criteria, and the center of minimal exponent) are asserted at a high level without any explicit statements of the required extension lemmas for twisted Hodge modules, without definitions of the two-parameter family, and without even schematic proof outlines; this prevents any check that the standard V-filtration properties extend as needed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. We respond to the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claims (vanishing theorems, restriction theorems, nontriviality criteria, and the center of minimal exponent) are asserted at a high level without any explicit statements of the required extension lemmas for twisted Hodge modules, without definitions of the two-parameter family, and without even schematic proof outlines; this prevents any check that the standard V-filtration properties extend as needed.

    Authors: We agree that the abstract, as written, presents the results at a high level without explicit definitions or references to the technical tools. To address this, we will revise the abstract to include a brief definition of the two-parameter family (Hodge level and rational parameter) and to indicate the role of twisted Hodge modules and V-filtrations in the construction. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract presents higher multiplier ideals as a new definition constructed from the pre-existing theory of mixed Hodge modules and V-filtrations on complex manifolds. It recovers the standard multiplier ideals at Hodge level zero and states that vanishing, restriction, and nontriviality properties follow from this framework, with applications to theta divisor singularities. No equations, fitted parameters, self-citations, or ansatzes are provided in the available text, and the construction is explicitly positioned as resting on independent prior theory rather than reducing to its own inputs by definition or renaming. The derivation chain is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The construction rests on the established theory of mixed Hodge modules and V-filtrations; the new objects are the higher multiplier ideals themselves.

axioms (1)
  • standard math Standard properties of mixed Hodge modules and V-filtrations along effective divisors on complex manifolds hold and extend to the twisted setting used here.
    The definition and all subsequent theorems are built directly on these background results from the literature.
invented entities (2)
  • higher multiplier ideals no independent evidence
    purpose: Two-parameter family of ideal sheaves generalizing classical multiplier ideals via Hodge level.
    Newly defined in the paper using mixed Hodge modules.
  • center of minimal exponent no independent evidence
    purpose: Generalization of the minimal log canonical center.
    New notion introduced to study the higher multiplier ideals.

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Forward citations

Cited by 1 Pith paper

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