Hodge theory of secant varieties

TL;DR

This paper investigates the Hodge-theoretic properties of secant varieties of smooth projective varieties, revealing connections to primitive cohomology and computing various Hodge invariants, thus refining existing results.

Contribution

It introduces new methods to compute Hodge and local cohomology invariants of secant varieties, removing previous positivity restrictions.

Findings

Local cohomological defect relates to primitive cohomology for secant lines.

The defect is zero for higher secant varieties of curves.

Computed Hodge-Lyubeznik numbers and refined cohomological structures.

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.