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Local Cohomological Defect and a Conjecture of Mustata-Popa

Andrew Burke

TL;DR

The paper develops a depth-driven framework for the Du Bois complexes to bound the local cohomological defect lcdef(X) and to prove a Musta\c{t}\u{a}-Popa conjecture. It shows lcdef(X) = n - min_{0≤p≤n}(\mathrm{depth} \underline{\Omega}_X^p + p) and proves a key transfer: if \mathrm{depth} \underline{\Omega}_X^p + p ≥ k for all p ≤ m, then the same holds for p ≥ k-m-2, with broader consequences expressed via perversity defects pdef(X,d). The approach relies on the theory of mixed Hodge modules and vanishing for graded de Rham pieces, yielding a topological–holomorphic equivalence for lcdef and enabling a chain of corollaries that recover and refine Dao–Takagi and Park–Popa results, including explicit bounds and criteria in terms of punctured-neighborhood cohomology. Together, these results provide precise control over lcdef through depths of Du Bois complexes and introduce the perversity-defect machinery to sharpen existing singularity invariants. The work thus connects Hodge-theoretic, topological, and perverse-constructible data to produce practical criteria for assessing singularities in complex algebraic varieties.

Abstract

We prove a general result on the depth of Du Bois complexes of a singular variety. We apply it to prove a conjecture of Mustata-Popa and to study the local cohomological defect, extending results of Ogus and Dao-Takagi over the complex numbers.

Local Cohomological Defect and a Conjecture of Mustata-Popa

TL;DR

The paper develops a depth-driven framework for the Du Bois complexes to bound the local cohomological defect lcdef(X) and to prove a Musta\c{t}\u{a}-Popa conjecture. It shows lcdef(X) = n - min_{0≤p≤n}(\mathrm{depth} \underline{\Omega}_X^p + p) and proves a key transfer: if \mathrm{depth} \underline{\Omega}_X^p + p ≥ k for all p ≤ m, then the same holds for p ≥ k-m-2, with broader consequences expressed via perversity defects pdef(X,d). The approach relies on the theory of mixed Hodge modules and vanishing for graded de Rham pieces, yielding a topological–holomorphic equivalence for lcdef and enabling a chain of corollaries that recover and refine Dao–Takagi and Park–Popa results, including explicit bounds and criteria in terms of punctured-neighborhood cohomology. Together, these results provide precise control over lcdef through depths of Du Bois complexes and introduce the perversity-defect machinery to sharpen existing singularity invariants. The work thus connects Hodge-theoretic, topological, and perverse-constructible data to produce practical criteria for assessing singularities in complex algebraic varieties.

Abstract

We prove a general result on the depth of Du Bois complexes of a singular variety. We apply it to prove a conjecture of Mustata-Popa and to study the local cohomological defect, extending results of Ogus and Dao-Takagi over the complex numbers.
Paper Structure (12 sections, 27 theorems, 73 equations)

This paper contains 12 sections, 27 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mixed Hodge modules
  4. Trivial Hodge module
  5. Du Bois Complexes
  6. Depth and local cohomological defect
  7. Perversity functions and perversity defect
  8. Depth and Local Cohomological Defect
  9. Fundamental Lemmas
  10. Applications
  11. Perversity Defect
  12. Perversity Defect

Key Result

Theorem 1

Let $k$ be an integer. Then we have $\mathop{\mathrm{lcdef}}\nolimits (X) \leq n-k$ if and only if $\mathop{\mathrm{depth}}\nolimits \underline{\Omega}_X^p + p \geq k$ for all $p \leq \lceil (k-3)/2 \rceil$.

Theorems & Definitions (53)

  • Theorem 1
  • Theorem 2
  • Conjecture 3
  • Corollary 4
  • Theorem 5
  • Theorem 6
  • Lemma 2.2
  • Lemma 3.1
  • Lemma 4.2
  • Definition 5.1
  • ...and 43 more