The proposal concerns the study of singularities of algebraic varieties. One studies these singularities in two settings: in characteristic zero and in positive characteristic. In each setting one defines invariants that measure the singularities. Despite the different definitions, the two sets of invariants exhibit many common features and subtle connections. In characteristic zero, the invariants to be studied (more precisely, the log canonical threshold, and the minimal log discrepancies) are defined in terms of divisorial valuations. A fundamental result, the existence of resolution of singularities, gives a strong finiteness statement when computing these invariants. The motivation for the interest in such invariants comes from their applications to birational geometry: certain conjectural properties of these invariants imply the termination of flips, one of the outstanding open problems in higher dimensions. The PI plans to study questions related to the main conjecture on log canonical thresholds, that asserts that in fixed dimension, there are no increasing sequences of such invariants. Another part of the proposal concerns invariants defined in positive characteristic via the Frobenius morphism. They come out of tight closure theory, but have many subtle connections with the invariants in characteristic zero. In particular, a basic conjecture in the field connects in a precise way the invariants in characteristic zero to those in positive characteristic, via reduction mod p. The PI proposes to use D-module theoretic ideas in order to study this conjecture.
Singularities are responsible for many phenomena, both in geometry and in arithmetic. For example, they govern the asymptotic of the number of solutions of congruences modulo prime powers, or the integrability of powers of polynomials. It has been realized in the past few years that invariants of singularities that show up in different contexts have deep connections amongst them. One can hope that by approaching singularities from these various points of view, one can get a richer picture, that can be used to attack problems with applications in other fields.
This project was concerned with invariants of singularities of algebraic varieties. Such invariants play a fundamental role in the classification of higher-dimensional varieties. Deep connections have been recently uncovered between such invariants that arise in characteristic zero, and invariants of singularities in positive characteristic, that first came up in commutative algebra. The PI has worked on several problems in this area. Together with Tommaso de Fernex and Lawrence Ein, he solved a special case of a conjecture of Shokurov, predicting that there are no strictly increasing sequences of certain invariants of singularities, called log canonical thresholds. The conjecture is related to one of the main open problems in birational geometry (Termination of Flips), that predicts that there are no infinite sequences consisting of a certain type of transformations. In a different direction, the PI worked with Vasudevan Srinivas on the connection between invariants of singularities in characteristic zero, and similar invariants in positive characteristic. There is a precise conjecture in this setting that predicts a relation between two sets of such invariants. It was long understood that this conjecture has arithmetic content, but this work managed to make this content precise. It showed that the conjecture is equivalent with another conjecture, this time only concerning smooth projective varieties. The latter conjecture is a statement in arithmetic geometry that is widely expected to hold (though it seems to be quite difficult to handle with present techniques). In joint work with Mattias Jonsson, the PI has worked on asymptotic versions of some invariants of singularities. The goal was to provide an algebraic framework to attack an open problem about a certain class of functions that appear in the analytic context (the Openness Conjecture on plurisubharmonic functions). They proposed an algebraic question, and showed that a positive answer would give a proof of the Openness Conjecture. The question was solved affirmatively in the two-dimensional case.
