Title: Local cohomology, Hodge theory and inversion of adjunction
Abstract: Local cohomology is a fundamental tool in commutative algebra and algebraic geometry. Over the complex numbers, the local cohomology of a smooth variety along a subvariety admits an action by differential operators and has an associated Hodge and weight filtration (due to M. Saito). These filtrations contain important singularity information about the subvariety (as evidenced by Mustațǎ-Popa's Hodge ideals). Mustațǎ-Popa also showed that local cohomology can detect Du Bois and rational singularities. I will explain how this point of view gives a new perspective on inversion of adjunction for such singularities, as well as their higher analogues in the LCI setting, based on joint work with Qianyu Chen and Sebastián Olano.