VECTOR BUNDLES ON MANIFOLDS

在这个项目中，我们从以下几个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，3)数论方法。在文[1]中，我们得到了(1)射影空间P^n(n*4)上的秩2-丛分裂为线丛的一个充要条件，这为关于P^n上向量丛分裂的重要猜想提供了一种新的方法；(2)通过阐明边界因子Y的数值有效性，我们对仿射C^3空间的非射影Moishezon紧化(X，Y)进行了分类，并证明了非Stein流形上的C^1-纤维空间的Steiness与S上某一秩2-丛的平凡性之间的等价性；(3)在某些条件下，我们证明了第一类Chern类，证明了以椭圆曲线为纤维的曲面上丛模的合理性，并确定了椭圆曲面上丛模的Picard群和阿尔巴ese映射。在2)中，我们学习…(1)半单李代数的最高权模，特别是紧Hermite对称空间上的最高权模；(2)在一般类型的Flag流形上引入了Radon变换的定义，找出了丛在Radon变换中的应用；(3)利用扭曲上同调群与外积的交集理论，得到了关于广义超几何函数的一个对偶。在文[3]中，(1)给出了L函数的傅立叶系数的积分表示式，它是正交群上的模形式的半整权尖形的提升，并在某些技术条件下证明了亚纯的连续性和函数方程，这些条件可以看作是Kohnen-Skoruppa关于二次Siegel尖形的结果的推广；(2)我们证明了局部域上具有正特征的p-进Galoi表示范畴与具有Frobenius映射的etale微分模范畴是等价的，特别是惯性群因子的有限表示对应于超收敛模的范畴。(3)研究了复流形情况下类似Riemann-Hilbert对应的p-进Hodge理论的Sheaficatin及其相关性。较少

In this project, we studied vector bundles on manifolds from the following various points of view : 1)algebraic geometric method, 2)algebraic analytic method, 3)number theoretic method. In 1), we obtained (1) a necessary and sufficient condition for a rank 2 bundle on projective space P^n (n*4) to split into line bundles which gives us a new approach to important conjectures concerning splitting of vector bundles on P^n, (2) we classified Non-projective Moishezon compactifications (X,Y) of affine C^3 space by clarifing numerically effectiveness of the boundary divisor Y and moreover, showed an equivalence between the Stein ess of a C^1- fiber space over a non-Stein manifold S and the triviality of certain rank 2 bundle on S., (3) we proved under some conditions on the first Chern class, the rationality of the moduli of bundles on surfaces with elliptic curves as fibers and have determined the Picard group and Albanese map of the moduli of bundles on elliptic surfaces. In 2), we studied … More (1) heighest weight modules of semi-simple Lie algebras, especially those corresponding to compact Hermitian symmetric spaces, (2) we have introduced the definition of Radon transformation on Flag manifolds of general type and founed usefulness of bundles to study the Radon transformation, (3) we got a duality concerning generalized hypergeometric functions by using the intersection theory on twisted cohomology group and the exterior products. In 3), (1) we gave an expression by integrals in terms of their Fourier coefficients of the L-functions which are liftings of cusp forms with haif integer weights to the modular forms on orthogonal groups and proved the meromorphic continuation and the functional equation under some technical conditions which can be viewed as a generalization of Kohnen-Skoruppa's result on quadratic Siegel cusp forms, (2) we showed an categorical equivalence between the category of p-adic Galoi representations over local fields with positive characteristic and the category of etale differential modules with Frobenius map and in particular, the ones whose representation of inertial group factors through finite representations correspond to overconvergent modules, (3) investigated sheaficatin of p-adic Hodge theory and their relativeness, which are analogue to Riemann-Hilbert correspondence in the case of complex manifolds. Less