Partial Cohomologically Complete Intersections via Hodge Theory

TL;DR

This paper explores a generalization of cohomologically complete intersections using Hodge theory, relating it to local cohomology, Du Bois complexes, and intersection cohomology, with applications to cones over rational homology manifolds.

Contribution

It introduces a new perspective on cohomologically complete intersections via Hodge modules and establishes inequalities and characterizations involving Hodge filtration and local cohomology.

Findings

Established a relation between the generalized property and Hodge filtration on local cohomology.

Derived inequalities on the codimension of non-perverse loci of shifted constant sheaves.

Described higher local cohomology modules of cones over projective rational homology manifolds.

Abstract

Using Saito's theory of mixed Hodge modules, we study a generalization of Hellus-Schenzel's "cohomologically complete intersection" property. This property is equivalent to perversity of the shifted constant sheaf. We relate the generalized version to the Hodge filtration on local cohomology, depth of Du Bois complexes, Hodge-Lyubeznik numbers and prove a striking inequality on the codimension of the non-perverse locus of the shifted constant sheaf. We study the case of cones over projective rational homology manifolds. We study when such varieties satisfy the weakened condition mentioned above as well as the partial Poincar\'{e} duality. To do this, we completely describe their higher local cohomology modules in terms of the Hodge theory of the corresponding projective variety. We apply this to the study of Hodge-Lyubeznik numbers and the intersection cohomology.