Microlocal Analysis and Complex Geometry

点击翻译按钮获取中文摘要

The question of using waves or particles to describe physical phenomenons like the propagation of fluids, gas, electricity and light has been a central issue of science since the beginning of scientific times. Microlocal Analysis develops a very geometrical manner of dealing with asymptotic calculus, econciling asymptotically in a remarkable manner the techniques of wave analysis (Fresnel's description of light) and of particle analysis (geometrical optics). On the other hand, Complex Geometry studies complex manifolds such solutions of polynomial equation in the projective space. The subject is on the crossroad of algebraic and differential geometry and has many applications in string theory. The purpose of this project is to apply techniques from microlocal analysis to problems arising from complex geometry, some of which are also interesting from physical point of view (Morse inequalities, asymptotic of Toeplitz operators, geometric quantization). Concretely, the project deals with semiclassical asymptotics for large N of Bergman and Szegö kernels on N-tensor powers of a holomorphic line bundle. These kernels are reproducing kernels of projectors on spaces of holomorphic sections. In physics terms, they are the projectors on the lowest Landau level of a particle in a magnetic field. The Bergman kernel is the density of states, describing a mixed state in which each ground state appears with equal weight, describing the zero temperature state of maximum entropy. The magnetic field is supposed to satisfy a Dirac quantization condition, i.e., it is the curvature of a line bundle. We then consider scaling up the magnetic field by the parameter N, hence we consider N-tensor powers of the bundle. The asymptotics of the kernels are expansions in powers of N, whose coefficients encode information about the curvature of the underlying manifold.For example, one goal is to settle the Ramadanov conjecture, to the effect that a hyper-surface in a complex manifold is equivalent to a sphere, if a certain (logarithmic) term in the expansion of its Szegö kernel vanishes. There are various physics interpretations of the asymptotics of Bergman kernel. One is to regard the underlying manifold as a phase space, and try to quantize it by using Toeplitz operators, following Berezin. One goal is to extend the Berezin-Toeplitz quantization to the case of line bundles endowed with singular Hermitian metrics, which appear naturally in algebraic geometry. Hence we could quantize a larger class of phase spaces than projective manifolds. Another goal is to establish the Berezin-Toeplitz quantization on general symplectic (non-Kähler) manifolds by using the projector on the spectral space of the low lying eigenvalues (bound states) of an appropriate Hamiltonian (Bochner-Laplacian). The zero sets of generic sections in this spectral space should be symplectic submanifolds, thus providing new structure theorems a la Donaldson.