Microlocal Analysis and Complex Geometry

自科学时代开始以来，使用波或粒子来描述流体、气体、电和光的传播等物理现象一直是科学的中心问题。微局部分析发展了一种非常几何的方式来处理渐近微积分，以一种引人注目的方式渐进地结合了波分析(菲涅耳对光的描述)和粒子分析(几何光学)的技术。另一方面，复几何研究射影空间中的复流形，如多项式方程的解。这门学科位于代数和微分几何的十字路口，在弦理论中有许多应用。这个项目的目的是将微局部分析技术应用到复杂几何问题中，其中一些问题从物理角度来看也很有趣(Morse不等式、Toeplitz算子的渐近性、几何量子化)。具体地说，该项目研究了大N的Bergman和Szegökernels关于全纯线丛的N张量幂的半经典渐近性。这些核是全纯截面空间上的射影的再生核。在物理学术语中，它们是磁场中粒子最低朗道能级上的投影仪。Bergman核是态的密度，描述了一个混合态，其中每个基态以相等的权重出现，描述了最大熵的零温态。假设磁场满足狄拉克量子化条件，即它是线丛的曲率。然后我们考虑用参数N来放大磁场，因此我们考虑了丛的N张量次方。核的渐近性是N的幂展开，其系数编码了关于基础流形的曲率的信息。例如，一个目标是解决Ramadanov猜想，如果在其Szegö核的展开式中的某个(对数)项消失，则复流形中的超曲面等价于球面。对伯格曼核的渐近性有不同的物理解释。一种是把基本流形看作一个相空间，在Berezin的基础上利用Toeplitz算子对其进行量子化。一个目标是将Berezin-Toeplitz量子化推广到具有奇异厄米度量的线丛的情况，这在代数几何中是自然出现的。因此，我们可以量子化比射影流形更大的一类相空间。另一个目标是在一般的辛流形(非Kähler流形)上建立Berezin-Toeplitz量子化，方法是利用适当的哈密顿量(Bochner-Laplace)的低位本征值(约束态)的谱空间上的投影器。这个谱空间中的一般截面的零集应该是辛子流形，从而提供了类似Donaldson的新的结构定理。

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.