This research concerns several projects in algebraic geometry. Classically, algebraic geometry is the study of solutions of systems of polynomial equations, and it plays a key role in numerous fields of mathematics, both pure and applied. The solution space of a system of polynomial equations has a very rich structure that can be smooth (continuous) or singular (discontinuous). This project is concerned with a new approach to singularities that arises from transferring ideas from complex geometry to an algebraic setting. The main goal is to further develop the theory of "Hodge ideals," which are subtle invariants of singularities.
In more detail, several directions of research will be pursued: extending the current version of Hodge ideals associated to reduced hypersurfaces (one would like extensions to Q-divisors or to ideals defining a subscheme that is reduced in codimension one); extending the study of Hodge filtrations on localizations at one element (which is equivalent to the study of Hodge ideals) to the study of Hodge filtrations on local cohomology modules of a regular local ring along an arbitrary ideal; investigating a potential analogue of Hodge ideals in positive characteristic; exploiting the connection between Hodge ideals and the motivic Chern transformation of Brasselet-Schuermann-Yokura; and developing tools based on more classical methods to give new proofs of results on Hodge ideals whose current proofs rely on Saito's theory of Hodge modules.
