About
One of my long-term interests is the study of invariants of singularities that come up in birational geometry, and more precisely, in dealing with linear systems on higher-dimensional algebraic varieties. These invariants can be numerical (such as the log canonical threshold or minimal log discrepancies) or given by ideals (such as the multiplier ideals). The log canonical threshold is particularly interesting since it comes up in various approaches to singularities: as the index of integrability of holomorphic functions, in connection with valuations, describing the asymptotic growth of the dimensions of jet schemes, as part of the spectrum of a singularity, as the negative of the largest root of the Bernstein-Sato polynomial, and others.
I have also been interested in connections between such invariants in characteristic zero and similar invariants defined in positive characteristic via the Frobenius morphism (one instance of such a connection is the relation between log canonical thresholds and F-pure thresholds via reduction mod p). Some of these connections are established by the work of people working in tight closure theory, while others are still conjectural, and related to deep conjectures in arithmetic geometry.
An interesting aspect is that the invariants and the techniques developed to study positive-characteristic singularities turned out to be useful also to prove global geometric statements, whose proofs in characteristic zero relies on vanishing theorems.
Other interests include toric varieties and birational geometry (especially in connection with asymptotic base loci and numerical invariants of line bundles and, more generally, of graded sequences of ideals).
