Partial Cohomologically Complete Intersections via Hodge Theory

TL;DR

The paper develops a comprehensive Hodge-theoretic framework to generalize the cohomologically complete intersection property via Saito's mixed Hodge modules. It proves that the invariant $c(X)$, defined from the Hodge filtration, equals the maximal $k$ for which depth$\left(\underline{\Omega}_X^p\right) \ge \dim X - p$ for all $p \le k$, and furnishes multiple equivalent characterizations using Hodge–Lyubeznik numbers and local cohomology, with the sharp inequality ${\rm HRH}(X) \le c(X)$ and criteria for equality. The work further derives powerful consequences for singularity invariants, including a bound on the generic local cohomological defect and a description of the non-perverse locus in terms of these Hodge-theoretic quantities. It applies the theory to cones over projective rational homology manifolds and to contractions of zero sections, giving explicit decompositions of dual objects and connections to the primitive cohomology, IH, and HL numbers. Finally, the results are specialized to determinantal varieties, where RW's framework yields precise computations of $\mathrm{lcdef}_{\rm gen}^{>0}$ and $c(Z_p)$, as well as explicit formulas for Hodge–Lyubeznik numbers in terms of primitive Hodge data. Overall, the paper builds a bridge between mixed Hodge theory, perverse/sheaf-theoretic invariants, and concrete singularity invariants with broad computational consequences.

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 Poincaré 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.

Theorems & Definitions (90)

• Theorem 1
• Theorem 2
• Theorem 3
• Remark 1.1
• Remark 1.2
• Example 1.3
• Theorem 4
• Remark 1.4
• Corollary 5
• Corollary 6