Geometry of analytic and algebraic varieties

This research project is in the field of algebraic geometry, one of the core areas of pure mathematics whose roots date back to the ancient Greeks. At its heart, algebraic geometry studies the geometry of the solution of polynomial equations. These are the simplest possible equations and so, not surprisingly, they play an important role in almost any scientific discipline. In particular algebraic geometry has close ties with differential and analytic geometry, commutative algebra, topology, number theory, physics, theoretical computer science, cryptography, and many areas of applied mathematics. The most fundamental problem in algebraic geometry is to classify all geometric objects defined by polynomial equations. The minimal model program is the most successful approach to this classification program and has recently had extraordinary success in classifying complex varieties, i.e. solution sets consisting of complex numbers. It is hoped that these techniques will extend to other contexts such as varieties defined over fields of positive characteristics and to complex analytic varieties. The PI will involve graduate students and post-docs in various aspects of this project. This project will generalize the results of the minimal model program over the complex numbers to the case of varieties over algebraically closed fields of positive characteristic (and to the case of mixed characteristics) as well as the case of Kahler varieties. In particular, the PI will show that the minimal model program holds for compact Kahler klt threefolds and fourfolds and will develop the theory of generalized klt Kahler varieties. The PI will investigate the minimal model program for threefolds in characteristic 3 and the pluricanonical maps of projective 3-folds over algebraically closed fields of large characteristics. The minimal model program in mixed characteristic will also be developed. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.