Hodge theory of secant varieties
Qianyu Chen, Bradley Dirks, Sebastian Olano, Debaditya Raychaudhury
TL;DR
This work develops a Hodge-theoretic framework for secant varieties, relating local cohomology defects to primitive cohomology and establishing weight filtrations, Hodge–Lyubeznik numbers, and intersection cohomology structures for secant varieties of lines and higher secant varieties of curves. It employs Saito’s theory of mixed Hodge modules to generalize and sharpen prior results under minimal positivity (3-very ampleness) and to remove restrictive positivity hypotheses. Concrete descriptions are given for the generation level of the Hodge filtration, the singular and intersection cohomology, and the $f Q$-factoriality defect, along with a Cappell–Shaneson-type equality between Hirzebruch and L-classes for these varieties. The results yield explicit criteria and formulas tying singularity properties to cohomological invariants, and they extend known results from curves to higher-dimensional varieties, including a thorough treatment of the Secant varieties of rational normal curves and higher secants of curves.
Abstract
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.