The proposer aims to study various aspects of algebraic varieties. The first part intends to establish a geometrically meaningful compactification of the moduli of algebraic varieties, especially for varieties of general type. A key problem is to understand the appropriate local deformation theory for varieties whose canonical sheaf is not locally free. The other parts focus on various special classes of algebraic varieties. Rationally connected varieties are the simplest algebraic varieties from many points of view, but their behavior over finite fields is still very poorly understood. The proposer aims to find on them rational curves defined over finite fields. As an extension of his earlier work on circle actions on links of singularities, the proposer will study how to compute the integral homology of links by giving a geometric explanation to the conjectures of Orlik.
In order to specify a sphere, one just needs to know 1 number: its radius. For an ellipsoid, one needs to specify 3 numbers: the lengths of the 3 semi axes. The theory of moduli of algebraic varieties aims to establish a similar pattern for more complicated geometric objects that can be described by algebraic equations. The proposer aims to study this question in general. The first question is to describe the necessary parameters, how many one needs and how these parameters relate to the underlying geometry. The main part of the proposed research aims to understand the geometric transitions that occur when one or more of the parameters become very large. For many applications, these are the most interesting phenomena related to moduli theory.
