Lyubeznik numbers for Pfaffian rings

Abstract

We study the structure of local cohomology with support in Pfaffian varieties as a module over the Weyl algebra DX of differential operators on the space of skew-symmetric matrices X=⋀2Cn. The simple composition factors of these modules are known by the work of Raicu-Weyman, 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 origin of the indecomposable summands referred to above.