Abstract:

Du Bois and rational singularities are two classes of singularities that show up in some of the most fundamental topics in algebraic geometry, such as the Minimal Model Program, and moduli theory. Over the last few years, there has been a lot of interest in their natural higher analogues, called k-Du Bois and k-rational singularities. These singularities have been extensively studied in the case of local complete intersection (lci) varieties but not much is known outside the lci case. In this thesis, we prove some of the first results in this field beyond the lci setting, with the main theorem being a higher analogue of a classical result of Kovács, i.e. we prove that the class of varieties with k-Du Bois singularities contains the class of varieties with k-rational singularities, which extends prior results of Mustaţă-Popa and Friedman-Laza in the lci setting.

The second part of the thesis concerns toric varieties, which are a well studied class of algebraic varieties admitting alternate descriptions in terms of convex geometric objects in \mathbb{R}^n. We prove new local vanishing results on toric varieties and build on the techniques therein to give a precise formula relating the graded de Rham complex of the intersection cohomology Hodge module \IC^H_X to the stalks of the intersection cohomology perverse sheaf \IC_X, when X is a toric variety.