Using Mixed Hodge Modules to Study Singularities

Abstract

Morihiko Saito’s theory of Hodge modules have made an incredible impact in the study of singularities. So far, the strongest results have been obtained in the case of hypersurface singularities, using the strong properties of the V -filtration along hypersurfaces which is built into the theory of Hodge modules. This thesis extends two important tools from the case of hypersurfaces. The first is a compatibility property between the Hodge filtration of a mixed Hodge module and the V - filtration along a higher codimension subvariety. The second is a formula explaining how to restrict to a smooth subvariety of higher codimension using the V -filtration along that subvariety. The main tool at work in proving these theorems is the blow-up along the smooth subvariety. There are two main applications of these theorems: the first is to analyze the Hodge and weight filtration on the local cohomology module along a singular locally complete intersection subvariety. We define the minimal exponent of a locally complete intersection variety and show that its value dictates when the Hodge filtration on local cohomology is equal to the pole order filtration. This shows that the minimal exponent understands information about k-du Bois singularities, and it turns out that the minimal exponent also understands k-rational singularities, by its relation to the weight filtration on local cohomology. The second application is to the study of the Fourier-Laplace transform of monodromic mixed Hodge modules. These modules naturally arise through Verdier’s specialization construction. We explicitly write out the Hodge and weight filtrations for such modules.