arXiv subjects

Bradley Dirks

Publications and source records attributed to Bradley Dirks.

At least 19 recordsLinked to original sources

arXiv2026

Partial Cohomologically Complete Intersections via Hodge Theory

We introduce an invariant $c(X)$ associated to any complex algebraic variety $X$, which for varieties with isolated singularities measures the failure of dual Kodaira-Akizuki-Nakano vanishing. In general, it is characterized by Hodge--Lyubeznik numbers, the depth of Du Bois complexes, and the Hodge filtration on local cohomology modules. We show that this invariant is computable in many examples, such as cones over rational homology manifolds or determinantal varieties.

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arXiv2026

A Hodge module construction of relative Du Bois complexes

We construct relative Du Bois and relative intersection Du Bois complexes for smooth-factorizable morphisms to a smooth complex base using relative de Rham complexes of mixed Hodge modules and Grothendieck duality. For a smooth morphism, these complexes recover the relative Kähler differentials. Comparisons with the relative Du Bois complexes of Kovács and Ning remain open. We construct finite filtrations on the absolute Du Bois and intersection Du Bois complexes whose graded pieces are the corresponding relative complexes tensored with differential forms from the base. We also study restriction to fibers and its relation to symbolic $m$-Du Bois and weak $m$-rational singularities.

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arXiv2026

Filtrations on Local Cohomology and Injectivity Theorems

We prove that the Hodge filtration on local cohomology is contained in the symbolic Ext filtration, and hence in the usual Ext filtration, answering a question of Mustata and Popa. Our main observation is that the Ext filtration admits a filtered-duality interpretation via Hartshorne's algebraic de Rham complex and its infinitesimal filtration. Comparing this construction with the filtered Du Bois complex yields the containment. The same mechanism proves the higher injectivity conjecture of Popa--Shen--Vo and several variants.

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arXiv2026

$F$-nilpotence and Hodge filtrations beyond isolated singularities

We give Hodge-theoretic characterizations of characteristic-zero varieties of open $F$-nilpotent and open weakly $F$-nilpotent type, extending a result of Srinivas and Takagi beyond isolated singularities. We prove one direction unconditionally and the converse assuming the weak ordinarity conjecture.

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arXiv2026

A Hodge Theoretic generalization of $\mathbb{Q}$-Homology Manifolds I: General Case

We study a natural Hodge-theoretic generalization of rational (or $\mathbb{Q}$-)homology manifolds through an invariant $\HRH(Z)$ attached to a complex algebraic variety $Z$. 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 $\HRH(Z)$ 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. We also introduce the generic local cohomological defect ${\rm lcdef}_{\textrm{gen}}(Z)$ and relate it to $\HRH(Z)$. Various examples are discussed at the end.

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arXiv2026

A Hodge Theoretic Generalization of $\mathbb{Q}$-Homology Manifolds II: Local Complete Intersections

Recently, the authors introduced and studied a singularity invariant of a complex algebraic variety $Z$, written $\HRH(Z)$ (for ``Hodge rational homology'' manifold level). In this paper, we focus on local complete intersection subvarieties. We relate $\HRH(Z)$ to various well-known invariants, like Bernstein--Sato polynomials and the Dimca-Maisonobe-Saito spectrum. In the hypersurface case it turns out that $\HRH(Z)$ can be completely characterized by these invariants, though higher codimension case is more subtle.

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arXiv2026

Microlocal Bernstein--Sato polynomials on singular ambient varieties

We introduce the microlocal Bernstein--Sato polynomial of a function on a possibly singular ambient variety, extending the theory of Saito. We show that, contrary to the smooth ambient setting, these polynomials are not generally equal to the reduced $b$-functions obtained by removing the trivial root. We define the minimal exponent and use it to study the singularities of the divisor and the Hodge filtration on local cohomology. Our main results include a generalization of Saito's theorem relating the minimal exponent to rational singularities, a characterization of purity of local cohomology, a Thom--Sebastiani formula for the minimal exponent, and a linear combination formula for Bernstein--Sato polynomials of ideals. When the ambient variety is a complete intersection with rational singularities, we provide effective algorithms for these Bernstein--Sato polynomials and implement them in Macaulay2.

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arXiv2026

Restrictions of mixed Hodge modules using generalized V-filtrations

We study generalized $V$-filtrations, defined by Sabbah, on $\mathcal D$-modules underlying mixed Hodge modules on $X\times \mathbf A^r$. Using cyclic covers, we compare these filtrations to the usual $V$-filtration, which is better understood. The main result shows that these filtrations can be used to compute the restriction functors $σ^!, σ^*$, where $σ\colon X \times \{0\} \to X \times \mathbf A^r$ is the inclusion of the zero section. As an application, we use the restriction result to study singularities of complete intersection subvarieties. These filtrations can be used to study the local cohomology mixed Hodge module. In particular, we classify when weighted homogeneous isolated complete intersection singularities in $\mathbf A^n$ are $k$-Du Bois and $k$-rational.

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arXiv2026

Characterization and finite descent of local cohomological invariants

We provide simple ``left-inverse characterizations'' of the recently introduced singularity invariants $c(Z)$, $w(Z)$, and ${\rm HRH}(Z)$ of an equidimensional variety $Z$. Combining this with a trace morphism, we establish descent results of these invariants for finite surjective morphisms.

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arXiv2026

Hodge theory of secant varieties

We study the local cohomology modules for the secant variety of lines of a smooth projective variety $Y$ and for higher secant varieties of smooth projective curves. We show that the local cohomological defect in the first case is related to the primitive cohomology of $Y$, and in the second case it is $0$. As applications, we compute their (intersection) Hodge-Lyubeznik numbers, the mixed Hodge structure on their singular cohomology, the pure Hodge structure on their intersection cohomology, the generating level of the Hodge filtration on their local cohomology modules and their $\mathbf Q$-factoriality defect. As byproducts, we recover and refine various results from the literature by removing restrictive positivity assumptions.

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arXiv2025

Multiplier modules, $V$-filtrations and Bernstein-Sato polynomials on singular ambient varieties

We show that the relation between multiplier ideals and $V$-filtration on the structure sheaf due to Budur-Mustaţă-Saito generalizes to singular irreducible varieties, by replacing multiplier ideals with multiplier modules and the structure sheaf with the intersection complex Hodge module. This is applied to a Skoda theorem for such modules as well as a $\mathcal D$-module theoretic proof of Ajit's formula relating the multiplier modules of an ideal to those of the Rees parameter in the extended Rees algebra. Moreover, we define a Bernstein-Sato polynomial for the pair of a variety and an ideal sheaf on it. We relate the roots to the jumping numbers of the multiplier modules. If the ideal is generated by a regular sequence on a rational homology manifold, we show that the absence of integer roots of the polynomial implies that the subvariety defined by the ideal is a rational homology manifold.

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arXiv2025

The minimal exponent of cones over smooth complete intersection projective varieties

We compute the minimal exponent of the affine cone over a complete intersection of smooth projective hypersurfaces intersecting transversely. The upper bound for the minimal exponent is proved, more generally, in the weighted homogeneous setting, while the lower bound is deduced from a general lower bound in terms of a strong factorizing resolution in the sense of Bravo and Villamayor.

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arXiv2025

Hirzebruch-Milnor classes of local complete intersections, minimal exponent, and applications to higher singularities

In this paper we use the deformation to the normal cone and the corresponding Verdier-Saito specialization to define and study (spectral) Hirzebruch-Milnor type homology characteristic classes for local complete intersections. Our main results describe vanishing properties of these classes in relation to the minimal exponent. As applications, we show how Hirzebruch-Milnor classes of local complete intersections with a projective singular locus can be used to detect higher Du Bois and higher rational singularities.

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arXiv2024

Some applications of microlocalization for local complete intersection subvarieties

Saito's microlocalization construction has been used to great effect in understanding hypersurface singularities. In this paper, we introduce what we believe to be a suitable analogue of the microlocalization construction for local complete intersection subvarieties. As evidence, we relate our construction to Saito's in the codimension one case. Moreover, we use this construction to study various natural questions concerning the minimal exponent of LCI subvarieties. We show that the minimal exponent agrees with the smallest Bernstein-Sato root, which was expected to be true. We also show that, in the isolated complete intersection singularities case, the minimal exponent agrees with the smallest non-zero spectral number, as defined by Dimca, Maisonobe and Saito. As applications of these results, we prove constructibility of the minimal exponent along certain Whitney stratifications and we prove that the spectrum (hence, the minimal exponent) is constant in equisingular families of ICIS varieties, in the sense of Gaffney.

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arXiv2024

Fourier transform and Radon transform for mixed Hodge modules

We give a generalization to bi-filtered $\mathcal D$-modules underlying mixed Hodge modules of the relation between microlocalization along $f_1,...,f_r \in \mathcal O_X(X)$ and vanishing cycles along $g = \sum_{i=1}^r y_i f_i$. This leads to an interesting isomorphism between localization triangles. As an application, we use these results to compare the $k$-plane Radon transform and the Fourier-Laplace transform for mixed Hodge modules. This is then applied to the Hodge module structure of certain GKZ systems.

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arXiv2024

V-filtrations and minimal exponents for locally complete intersection singularities

We define and study a notion of minimal exponent for a locally complete intersection subscheme $Z$ of a smooth complex algebraic variety $X$, extending the invariant defined by Saito in the case of hypersurfaces. Our definition is in terms of the Kashiwara-Malgrange $V$-filtration associated to $Z$. We show that the minimal exponent describes how far the Hodge filtration and order filtration agree on the local cohomology $H^r_Z({\mathcal O}_X)$, where $r$ is the codimension of $Z$ in $X$. We also study its relation to the Bernstein-Sato polynomial of $Z$. Our main result describes the minimal exponent of a higher codimension subscheme in terms of the invariant associated to a suitable hypersurface; this allows proving the main properties of this invariant by reduction to the codimension $1$ case. A key ingredient for our main result is a description of the Kashiwara-Malgrange $V$-filtration associated to any ideal $(f_1,\ldots,f_r)$ in terms of the microlocal $V$-filtration associated to the hypersurface defined by $\sum_{i=1}^rf_iy_i$.

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arXiv2024

An introduction to $V$-filtrations

We give an introduction to the theory of $V$-filtrations of Malgrange and Kashiwara. After discussing the basic properties of this construction (in the case of a smooth hypersurface and, later, in the general case), we describe the connection with the theory of $b$-functions. As an example, we treat the case of weighted homogeneous isolated singularities. We discuss the compatibility of $V$-filtrations with proper push-forward and duality and the connection with nearby and vanishing cycles via the Riemann-Hilbert correspondence. We end by describing some invariants of singularities via the $V$-filtration.

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arXiv2023

Verdier specialization and restrictions of Hodge modules

We give an explicit formula to express the cohomological pullback functors of Hodge modules under closed immersions of smooth varieties using Verdier specializations and $V$-filtrations of Kashiwara and Malgrange. This was locally obtained by the first two authors assuming the existence of global defining functions. We also give a quite simplified proof of the theorem reducing to the monodromical case via the Verdier specialization and using induction on codimension.

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