Hodge filtration on local cohomology, Du Bois complex and local cohomological dimension

Mustaţă, Mircea; Popa, Mihnea

doi:10.1017/fmp.2022.15

Forum of Mathematics, Pi

2022

DOI: 10.1017/fmp.2022.15

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Hodge filtration on local cohomology, Du Bois complex and local cohomological dimension

Mircea Mustaţă

Mihnea Popa

Abstract: We study the Hodge filtration on the local cohomology sheaves of a smooth complex algebraic variety along a closed subscheme Z in terms of log resolutions and derive applications regarding the local cohomological dimension, the Du Bois complex, local vanishing and reflexive differentials associated to Z.

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Cited by 7 publications

References 51 publications

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The higher Du Bois and higher rational properties for isolated singularities

Friedman,

Laza

2023

J. Algebraic Geom.

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Higher rational and higher Du Bois singularities have recently been introduced as natural generalizations of the standard definitions of rational and Du Bois singularities. In this note, we discuss these properties for isolated singularities, especially in the locally complete intersection (lci) case. First, we reprove the fact that a k k -rational isolated singularity is k k -Du Bois without any lci assumption. For isolated lci singularities, we give a complete characterization of the k k -Du Bois and k k -rational singularities in terms of standard invariants of singularities. In particular, we show that k k -Du Bois singularities are ( k − 1 ) (k-1) -rational for isolated lci singularities. In the course of the proof, we establish some new relations between invariants of isolated lci singularities and show that many of these vanish. The methods also lead to a quick proof of an inversion of adjunction theorem in the isolated lci case. Finally, we discuss some results specific to the hypersurface case.

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The higher Du Bois and higher rational properties for isolated singularities

Friedman,

Laza

2023

J. Algebraic Geom.

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Mixed Hodge Structure on Local Cohomology with Support in Determinantal Varieties

Perlman¹

2023

International Mathematics Research Notices

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We employ the inductive structure of determinantal varieties to calculate the mixed Hodge module structure of local cohomology modules with determinantal support. We show that the weight of a simple composition factor is uniquely determined by its support and cohomological degree. As a consequence, we obtain the equivariant structure of the Hodge filtration on each local cohomology module. Finally, as an application, we provide a formula for the generation level of the Hodge filtration on these modules.

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Singularities of secant varieties from a Hodge theoretic perspective

Olano,

Raychaudhury,

Song

2024

Proceedings of London Math Soc

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We study the singularities of secant varieties of smooth projective varieties using methods from birational geometry when the embedding line bundle is sufficiently positive. More precisely, we study the Du Bois complex of secant varieties and its relationship with the sheaves of differential forms. Through this analysis, we give a necessary and sufficient condition for these varieties to have ‐Du Bois singularities (in a sense that was proposed in Shen, Venkatesh, and Vo [On k‐Du Bois and k‐rational singularities, arXiv e‐prints (June 2023), arXiv:2306.03977]). In addition, we show that the singularities of these varieties are never higher rational, by giving a classification of the cases when they are pre‐1‐rational. From these results, we deduce several consequences, including a Kodaira–Akizuki–Nakano type vanishing result for the reflexive differential forms of the secant varieties.

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Hodge filtration on local cohomology, Du Bois complex and local cohomological dimension

Cited by 7 publications

References 51 publications

The higher Du Bois and higher rational properties for isolated singularities

The higher Du Bois and higher rational properties for isolated singularities

Mixed Hodge Structure on Local Cohomology with Support in Determinantal Varieties

Singularities of secant varieties from a Hodge theoretic perspective

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