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A Hodge Theoretic generalization of $\mathbb{Q}$-Homology Manifolds

Bradley Dirks, Sebastian Olano, Debaditya Raychaudhury

TL;DR

We develop ${\rm HRH}(Z)$, a Hodge-theoretic generalization of rational homology manifolds for pure $d$-dimensional complex varieties, defined via the quasi-isomorphism behavior of the morphisms ${\phi^p}$ relating Du Bois and dual Du Bois complexes. The framework, built on Saito's mixed Hodge modules, connects ${\rm HRH}(Z)$ to local cohomology and the cohomology of links upon embedding $Z$ into a smooth ambient space, yielding a partial Poincaré duality that sharpens our understanding of higher singularities. In the local complete intersection setting, HRH is governed by a network of integer invariants, including $p(Q^\mathbb{Z},F)$, the spectrum ${\rm Sp}_{\min,\mathbb{Z}}(Z,x)$, and the Bernstein–Sato data; in hypersurfaces, HRH is completely determined by $p(Q^\mathbb{Z},F)$ via the relation ${HRH}(Z)=p(Q^\mathbb{Z},F)+n-1$, with a direct link to the Milnor spectrum. The work also introduces the generic local cohomological defect ${\rm lcdef}_{\rm gen}(Z)$ and proves sharp codimension bounds for the non-rationally smooth locus, refined via normal slices and hyperplane sections. Overall, the paper provides a cohesive, computable Hodge-theoretic toolkit to compare and quantify higher Du Bois vs higher rational singularities and to extract duality information from local geometric data, including isolated singularities via Milnor fiber spectra.

Abstract

We study a natural Hodge theoretic generalization of rational (or $\mathbb{Q}$-)homology manifolds through an invariant ${\rm HRH(Z)}$ where $Z$ is a complex algebraic variety. The defining property of this notion encodes the difference between higher Du Bois and higher rational singularities for local complete intersections, which are two classes of singularities that have recently gained much attention. We show that ${\rm HRH(Z)}$ can be characterized when the variety $Z$ is embedded into a smooth variety using the local cohomology mixed Hodge modules. Near a point, this is also characterized by the local cohomology of $Z$ at the point, and hence, by the cohomology of the link. We give an application to partial Poincaré duality. In the case of local complete intersection subvarieties, we relate ${\rm HRH(Z)}$ to various invariants. In the hypersurface case it turns out that ${\rm HRH(Z)}$ can be completely characterized by these invariants. However, for higher codimension subvarieties, the behavior is rather subtle, and in this case we relate ${\rm HRH(Z)}$ to these invariants through inequalities and give some conditions on when equality holds.

A Hodge Theoretic generalization of $\mathbb{Q}$-Homology Manifolds

TL;DR

We develop , a Hodge-theoretic generalization of rational homology manifolds for pure -dimensional complex varieties, defined via the quasi-isomorphism behavior of the morphisms relating Du Bois and dual Du Bois complexes. The framework, built on Saito's mixed Hodge modules, connects to local cohomology and the cohomology of links upon embedding into a smooth ambient space, yielding a partial Poincaré duality that sharpens our understanding of higher singularities. In the local complete intersection setting, HRH is governed by a network of integer invariants, including , the spectrum , and the Bernstein–Sato data; in hypersurfaces, HRH is completely determined by via the relation , with a direct link to the Milnor spectrum. The work also introduces the generic local cohomological defect and proves sharp codimension bounds for the non-rationally smooth locus, refined via normal slices and hyperplane sections. Overall, the paper provides a cohesive, computable Hodge-theoretic toolkit to compare and quantify higher Du Bois vs higher rational singularities and to extract duality information from local geometric data, including isolated singularities via Milnor fiber spectra.

Abstract

We study a natural Hodge theoretic generalization of rational (or -)homology manifolds through an invariant where is a complex algebraic variety. The defining property of this notion encodes the difference between higher Du Bois and higher rational singularities for local complete intersections, which are two classes of singularities that have recently gained much attention. We show that can be characterized when the variety is embedded into a smooth variety using the local cohomology mixed Hodge modules. Near a point, this is also characterized by the local cohomology of at the point, and hence, by the cohomology of the link. We give an application to partial Poincaré duality. In the case of local complete intersection subvarieties, we relate to various invariants. In the hypersurface case it turns out that can be completely characterized by these invariants. However, for higher codimension subvarieties, the behavior is rather subtle, and in this case we relate to these invariants through inequalities and give some conditions on when equality holds.
Paper Structure (22 sections, 73 theorems, 395 equations)

This paper contains 22 sections, 73 theorems, 395 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mixed Hodge modules
  4. V-filtration, specialization and spectrum
  5. Poincaré duality and higher singularities
  6. Hodge Rational Homology Manifold Level
  7. Definition and basic properties
  8. Embedded case
  9. Local cohomology at a point and link invariants
  10. Partial Poincaré duality
  11. Generic local cohomological defect
  12. Local Complete Intersection Case
  13. Hypersurfaces
  14. Reinterpretation using specialization
  15. Integer invariants and relations among them
  16. ...and 7 more sections

Key Result

Theorem 1

Let $Z$ be an embeddable complex algebraic variety with $\mathop{\mathrm{HRH}}\nolimits(Z) \geq k$. Then for all $i\in \mathbb{Z}$, we have isomorphisms

Theorems & Definitions (177)

  • Definition 1
  • Theorem 1
  • Remark 2
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • ...and 167 more