9701779 de Cataldo This project is concerned with two problems: (1) the study of singular hermitian metrics on vector bundles and (2) codimension two subvarieties of quadrics. For the first problem, the principal investigator will determine a theory of singular hermitian metrics for holomorphic vector bundles on complex manifolds. The aim is to develop a basic understanding of them with special regards to positivity. For the second problem, the principal investigator will study codimension two subvarieties of dimension at least five. He will work on the classification of low degree submanifolds; the classification of submanifolds which carry special structure; the role of vector bundles in the above classification; the size of the Hilbert scheme of codimension two submanifolds not of general type; the rationality and unirationality of the earlier mentioned varieties; and the adjunction theoretic properties of these submanifolds. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.
