On higher Du Bois and higher rational pairs

Abstract

Du Bois and rational singularities play a central role in birational geometry and the minimal model program, governing key vanishing and deformation properties of varieties with mild singularities. Motivated by Hodge theory, a recent line of work has introduced higher analogues of these classes, refining them through the filtered Du Bois complex. While this theory applies to varieties, there is no current literature addressing pairs -- the natural category for the minimal model program. This thesis unifies the two perspectives and develops the theory of higher Du Bois and higher rational singularities for pairs, in the sense of the minimal model program. We extend various foundational results in this setting, including Bertini-type theorems, stability under finite maps, and the implication that $m$-rational pairs are $m$-Du Bois. The central technical contribution is a generalized Kovács–Schwede injectivity theorem for pairs, which is of independent interest. Together, these provide a framework for higher singularities that is compatible with the minimal model program and opens the way to further applications in birational geometry and Hodge theory.