Complex Analysis and Geometry

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

Complex Analysis and Geometry The research proposed deals with several areas on the interface of complex analysis and algebraic or symplectic geometry. The main project outlined seeks to develop a Delzant theory to classify completely integrable systems on compact manifolds with real algebraic type singularities of their action variables. These are meant to include the systems arising from spherical varieties (Gelfand-Cetlin examples) and those from quiver varieties at a minimum. The key tool would be Bohr-Sommerfeld level sets and their interplay with the geometry of the moment map in Hamiltonian dynamics. Other topics to be treated include regularity of exterior Monge-Ampere solutions and their relation to pluripotential theory; asymptotic geometry at infinity of complex manifolds, and renormalized Chern classes on them; some cases of the Hodge conjecture related to special Hodge-classes on higher dimensional complex manifolds derived from K3 surfaces, and the algebraicization of certain Grauert tubes. Philosophically, the first project is based on an idea going back to the beginning of quantum mechanics: the Bohr-Sommerfeld levels of a mechanical system, the set of configurations of a classical mechanical system which would contribute significantly to a quantum mechanical understanding of the system, as in, e.g., spectral lines of atoms. There was a later dual description of mechanics in terms of wave functions, and relating the two has been a source of great insight in physics and mathematics. What we suspect we have found, and are trying to prove, is that this Bohr-Sommerfeld correspondence underlies certain classification questions in classical mechanical systems: how many integrable systems are there? The Bohr-Sommerfeld levels are related directly to classical loci on the underlying space of the mechanical system, while the corresponding wave functions allow one to read off the algebraic geometry of the system. The main problem proposed is to make this "correspondence" rigorous, to the point where the systems can be classified by the combinatorial relations among the Bohr-Sommerfeld loci, and some global algebraic geometric data.