On higher singularities and algebraic K-theory

Abstract

This thesis applies Hodge-theoretic methods to study singularities in algebraic geometry, and uses the findings to obtain new results in algebraic $K$-theory. A first contribution is the introduction of new notions of \textit{higher Du Bois and higher rational singularities} for general complex algebraic varieties, extending previous definitions and results in the local complete intersection (lci) setting. A second theme is a careful study of the cohomological properties of the \textit{Du Bois complexes}, which are objects that show up naturally in the study of higher singularities. Finally, we apply these Hodge-theoretic inputs to study homotopy invariance in algebraic $K$-theory. We apply vanishing theorems for the Du Bois complexes to relate higher Du Bois singularities to \textit{$K$-regularity}, a notion that measures the homotopy invariance of algebraic $K$-groups. As a consequence, we obtain explicit numerical conditions for $K$-regularity and provide new regularity criteria strengthening Vorst's conjecture for local complete intersections.