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VECTOR BUNDLES ON MANIFOLDS

流形上的矢量束

基本信息

批准号：
06640054
负责人：
SUMIHIRO Hideyasu
金额：
$ 1.34万
依托单位：
HIROSHIMA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1994
资助国家：
日本
起止时间：
1994 至 1995
项目状态：
已结题

项目摘要

In this project, we studied vector bundles on manifolds from the following various points of view. 1) algebraic geometric method., 2) algebraic analytic method to study infinite dimensional Lie algebras and quantum groups., 3) number theoretic method to study modular forms and L-functions on classical groups., 4) differential geometric method to study quantization of infinite dimensional Lie groups by Feynman path integrals. In 1), we obtained a necessary and sufficient condition for a rank 2 bundle on projective space P^n (n <greater than or equal> 4) to split into line bundles by researching algebro-geometric, differential geometric and topological properties of determinantal varieties associated to rank 2 bundles on P^n which gives us a new approach to important conjectures concerning splitting of vector bundles on P^n. In 2), we proved a character formula of Kazdan-Lusztig type for irreducible highest weight modules over affine Lie algebras with integral highest weight case and als … More o extended this result to the rational highest weight case by constructing a theory on Radon transformations of P^1 bundles. Moreover we generalized generic hypergeometric equations on Grassmann varieties to differential equations on Hermite symmetric spaces and researched relations between the above Radon transformations and a condition concerning their holonomicity. In 3), the L-functions which are liftings of cusp forms with half integer weights to modular forms on orthogonal groups were studied in detail and we gave the L-functions an expression by integrals in terms of their Fourier coefficients and proved the meromorphic continuation and the functional equation under some technical conditions which can be viewed as a generalization of the Kohnen-Skoruppa's result on quadratic Siegel cusp forms. In 4), we constructed representation modules of infinite dimensional Lie groups, e.g.Kac-Moody Lie groups by Feynman path integrals and studied integral operators on infinite dimensional manifolds geometrically. Less

在这个项目中，我们从以下不同的角度研究了流形上的向量丛。1)代数几何方法。，2)代数分析方法研究无限维李代数和量子群，3)数论方法研究经典群上的模形式和L-函数，4)微分几何方法研究量子化的无限维李群的费曼路径积分。在[1]中，通过研究射影空间P^n（n4）上秩为2的丛的行列式簇的<greater than or equal>代数几何、微分几何和拓扑性质，得到了射影空间P^n（n4）上秩为2的丛分裂为线丛的一个充要条件，为研究P^n上向量丛分裂的重要问题提供了新的途径。在2）中，我们证明了仿射李代数上不可约最高权模的一个Kazdan-Lusztig型特征标公式， ...更多信息 o通过构造P^1丛的Radon变换的理论，将这个结果推广到有理最高权的情形。此外，我们还将Grassmann簇上的一般超几何方程推广到Hermite对称空间上的微分方程，并研究了上述Radon变换与其完整性条件之间的关系。在3）中，我们详细研究了正交群上带半整数权的尖点型到模型的提升L-函数，给出了L-函数关于Fourier系数的积分表达式，并在一定的条件下证明了L-函数的亚纯延拓和函数方程，它可以看作是Kohnen-Skoruppa关于二次Siegel尖点型的结果的推广。在4）中，我们利用Feynman路径积分构造了无限维李群，如Kac-Moody李群的表示模，并几何地研究了无限维流形上的积分算子。少