Du Bois complexes and singularity theory

Abstract

In this dissertation, we study the Du Bois complexes of varieties over complex numbers, which are generalizations of K¨ahler differentials for singular varieties, and explore their applications in singularity theory. In the first part, we introduce new notions of m-Du Bois and m-rational singularities, extending the existing definitions in the case of local complete intersections (LCI), to include natural examples beyond this setting. These notions provide a gradual refinement of notions associated with singularities, offering a measure of how far a variety is from being smooth, ranging from Du Bois singularities to smoothness. We show that varieties with m-rational singularities are m-Du Bois, extending previous results of Mustat¸ă-Popa and Friedman-Laza in the LCI and the isolated singularities cases. We also study other properties of these singularities, such as partial Hodge symmetries. The second part of the dissertation develops generic vanishing theory in the singular setting. Using Du Bois complexes, we establish appropriate generic vanishing theorems for singular varieties, generalizing the well-known generic vanishing theorem by Green and Lazarsfeld and the generic vanishing theorem of Nakano type by Popa and Schnell. Our results clarify the limitations of the naive generalizations, which were pointed out to be false by Hacon and Kovács.