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Microlocal Analysis and Complex Geometry

微局部分析和复杂几何

基本信息

批准号：
188691369
负责人：
Professor Dr. George Teodor Marinescu
金额：
--
依托单位：
Abteilung Mathematik
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2010
资助国家：
德国
起止时间：
2009-12-31 至 2014-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/188691369?language=en
关键词：
Microlocal Analysis Complex Geometry

项目摘要

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.

自科学时代开始以来，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核心问题。微局部分析发展了一种非常几何的方式来处理渐近演算，以一种显着的方式渐近地处理波分析（菲涅耳对光的描述）和粒子分析（几何光学）的技术。另一方面，复几何研究复流形，如射影空间中多项式方程的解。这门学科是代数几何和微分几何的交叉点，在弦理论中有许多应用。这个项目的目的是应用技术从微局部分析所产生的问题，从复杂的几何，其中一些也是有趣的从物理的角度来看（莫尔斯不等式，渐近Toeplitz运营商，几何量化）。具体地说，该项目涉及全纯线丛的N-张量幂上的Bergman和Szegö核的大N的半经典渐近性。这些核是全纯截面空间上投影器的再生核。用物理学术语来说，它们是磁场中粒子最低朗道能级上的投影器。伯格曼核是状态密度，描述了一个混合状态，其中每个基态以相等的权重出现，描述了最大熵的零温度状态。假设磁场满足狄拉克量子化条件，即，它是线束的曲率。然后我们考虑通过参数N放大磁场，因此我们考虑丛的N-张量幂。核函数的渐近性是N的幂次展开，其系数编码了底层流形的曲率信息。例如，一个目标是解决Ramadanov猜想，即如果Szegö核函数展开中的某个（对数）项为零，则复流形中的超曲面等价于球面。Bergman核的渐近性有多种物理解释。一种是将底层流形视为相空间，并尝试使用Toeplitz算子，遵循Berezin。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，这在代数几何中自然出现。因此，我们可以量化比投影流形更大类的相空间。另一个目标是建立一般辛（非Kähler）流形上的Berezin-Toeplitz量子化，通过使用适当的Hamilton（Bochner-Laplacian）的低本征值（束缚态）的谱空间上的投影。在这个谱空间中一般截面的零点集应该是辛子流形，从而提供了类似于唐纳森的新的结构定理。