Lazarsfeld 9713149 This award supports work on a number of problems in algebraic geometry. The first is to explore the connections between invariants from higher dimensional complex geometry and the rich classical geometry of abelian varieties and Riemann surfaces. The principal investigator also proposes to continue his work on the geometry of irregular varieties, as well as on the problem of trying to find lower bounds on the local positivity of an ample line bundle at a very general point of a smooth projective variety of arbitrary dimension. Finally, he will investigate some questions growing out of a recent observation with Bo Ilic to the effect that if X is a projective variety of dimension n with non-negative cotangent bundle, then the least degree of a projective embedding of X must grow essentially exponentially in n. 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.