Lyubeznik numbers for Pfaffian rings

TL;DR

This paper investigates the structure of local cohomology modules supported on Pfaffian varieties, revealing their composition factors and computing Lyubeznik numbers for Pfaffian rings, especially distinguishing between odd and even matrix sizes.

Contribution

It provides a detailed analysis of local cohomology modules over Pfaffian varieties and explicitly computes Lyubeznik numbers for Pfaffian rings, highlighting differences based on matrix size parity.

Findings

Local cohomology modules are semi-simple when n is odd.

When n is even, modules decompose from the pole order filtration.

Lyubeznik numbers are explicitly determined for Pfaffian rings.

Abstract

We study the structure of local cohomology with support in Pfaffian varieties as a module over the Weyl algebra D_X of differential operators on the space X of n x n complex skew-symmetric matrices. The simple composition factors of these modules are known by the work of Raicu-Weyman in 2016, and when n is odd, the general theory implies that the local cohomology modules are semi-simple. When n is even, we show that the local cohomology is a direct sum of indecomposable modules coming from the pole order filtration of the Pfaffian hypersurface. We then determine the Lyubeznik numbers for Pfaffian rings by computing local cohomology with support in the homogeneous maximal ideal of the indecomposable summands referred to above.