Relative De Rham Complexes, Families of Varieties

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Kleiman Kovacs 9600089 This awards supports the research of Dr. S. Kovacs as a post-doctoral associate of Professor S. Kleiman to work on problems in complex algebraic geometry. He hopes to find a construction for the relative De Rham complex for non-smooth morphisms. He also hopes to study log flips of complex varieties and to try to confirm Kollar's conjecture that certain Abelian surfaces blown-up at a single point give cones which decompose into non-trivial finitely generated part and a non-trivial nowhere locally finitely generated part. This research is in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which blossomed to the point where it has, in the past 10 years, solved problems that have stood for centuries. Originally, it treated figures defined in the plane by the simplest of equations, namely polynomials. Today, the field uses methods not only from algebra, but also from analysis and topology, and conversely it is extensively used in those fields. Moreover it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics.