The proposed research is in the general field of algebraic geometry, and primarily focuses on the problem of extension of topological invariants from manifolds to singular varieties and on problems in birational geometry of higher dimensional varieties. The principal techniques involved in this research are those coming from the theories of motivic integration and singularities of pairs. First, de Fernex proposes to further develop the theory of motivic integration, and to extend the results of a recent article on stringy Chern classes of singular varieties that he coauthored with Lupercio, Nevins and Uribe. Portion of this project stems from a nice interpretation of this results in the context of Deligne-Mumford stacks. Another portion of this project addresses the formulation of a theory of motivic integration involving derived categories and perverse sheaves; one of the aims in mind is the unification of the theories of stringy Chern classes and elliptic genera of singular varieties. A second collection of projects addresses the study of the birational geometry of Fano varieties and Mori fiber spaces, and particular attention is devoted to questions regarding nonrationality and birational rigidity of these varieties. Two more projects are proposed by de Fernex. One of these deals with a characterization of ampleness of line bundles via asymptotic cohomological vanishings, and is joint work with K""uronya and Lazarsfeld. The other project addresses the question of extendibility, to a given ambient variety, of rational fibrations defined on subvarieties with ample normal bundle; this is in collaboration with Beltrametti and Lanteri.
The first main project proposed here is based on a fundamental theorem in algebraic geometry, Hironaka's ``resolution of singularities'', which in its simplest form states that every singular complex algebraic variety can be modified into a manifold without altering the locus where it is already nonsingular. The fact that associated to any singular variety there exists a nonsingular one which looks ``almost the same'' suggests the idea that one should be able to extend topological invariants from manifolds to singular varieties by just looking at the varieties after resolving their singularities. However the resolution of singularities is typically not unique, so one needs to proceed cautiously; it is at this point that motivic integration comes into play: essentially, it is the technical tool used here to ensure that things, if defined suitably, do not depend on the chosen resolution. Hironaka's theorem is also crucial in the study of the birational properties of algebraic varieties and, in particular, it is used in the part of the proposed research dealing with questions concerning their nonrationality and birational rigidity. Modern techniques, based on a delicate analysis of singularities through their resolutions and accurate quantitative estimates of their nastiness, are here employed to address these questions, some of which are in fact quite classical and still open.
