Title of Seminar: Algebraic Geometry Preprint Seminar
Title of Talk: The relative Du Bois complex
Speaker: Arnab Roy, TIFR, Mumbai
Date: February 27, 2024
Time: 16:00:00 Hours
Venue: AG-77
Abstract: For a smooth projective variety $X$ over complex numbers, one gets a Hodge structure(pure) on the singular cohomology of $X$ in complex coefficients. Which can be obtained by the spectral sequence associated to the algebraic De Rham complex of $X$. In case of nonsingular varieties the singular cohomology admits Deligne's Mixed Hodge structure and the Du Bois complex (introduced by P. Du Bois in 1981) is the complex whose associated spectral sequence recovers the mixed Hodge structure. In this sense the Du Bois complex is a generalization of the algebraic De Rham complex. The Du Bois complex is a filtered complex, whose graded parts are in $D^b_\mathrm{coh}(X)$, these graded pieces are interesting on their own, as noted in Steenbrink(1985) these pieces provide generalization of Kodaira type vanishing theorems for singular spaces. The preprint named 'The relative Du Bois complex -- on a question of S. Zucker' {https://arxiv.org/abs/2307.07192}by Sándor J Kovács, Behrouz Taji takes the first step to generalize the Du Bois complex for a family (More precisely they construct the relative Du Bois complex when the base is a smooth curve). Similar generalisation of the graded pieces was already done by the first named author in the paper "smooth families over rational and elliptic curves". I will discuss the construction of the relative complex following the preprint and try to motivate why one should care for such a construction, If time permits I would like to discuss some applications of the relative graded pieces.
Title of Talk: The relative Du Bois complex
Speaker: Arnab Roy, TIFR, Mumbai
Date: February 27, 2024
Time: 16:00:00 Hours
Venue: AG-77
Abstract: For a smooth projective variety $X$ over complex numbers, one gets a Hodge structure(pure) on the singular cohomology of $X$ in complex coefficients. Which can be obtained by the spectral sequence associated to the algebraic De Rham complex of $X$. In case of nonsingular varieties the singular cohomology admits Deligne's Mixed Hodge structure and the Du Bois complex (introduced by P. Du Bois in 1981) is the complex whose associated spectral sequence recovers the mixed Hodge structure. In this sense the Du Bois complex is a generalization of the algebraic De Rham complex. The Du Bois complex is a filtered complex, whose graded parts are in $D^b_\mathrm{coh}(X)$, these graded pieces are interesting on their own, as noted in Steenbrink(1985) these pieces provide generalization of Kodaira type vanishing theorems for singular spaces. The preprint named 'The relative Du Bois complex -- on a question of S. Zucker' {https://arxiv.org/abs/2307.07192}by Sándor J Kovács, Behrouz Taji takes the first step to generalize the Du Bois complex for a family (More precisely they construct the relative Du Bois complex when the base is a smooth curve). Similar generalisation of the graded pieces was already done by the first named author in the paper "smooth families over rational and elliptic curves". I will discuss the construction of the relative complex following the preprint and try to motivate why one should care for such a construction, If time permits I would like to discuss some applications of the relative graded pieces.