VECTOR BUNDLES ON MANIFOLDS

在这个项目中，我们从以下几个方面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[1]中，我们得到了（1）射影空间P^n（n*4）上秩为2的丛分裂为线丛的一个充要条件，这为研究P^n上向量丛分裂的重要问题提供了一个新的途径，（2）通过阐明边界因子Y的数值有效性，对仿射C^3空间的非射影Moishezon紧化（X，Y）进行了分类，证明了非Stein流形S上C^1-纤维空间的Stein ess与S上某个秩为2的丛的平凡性之间的等价性，(3)在第一类Chern类上，在一定条件下证明了以椭圆曲线为纤维的曲面上丛的模的合理性，并确定了椭圆曲面上丛的模的Picard群和Albanese映射.（2）我们研究 ...更多信息 (1)（2）在一般类型的Flag流形上引入了Radon变换的定义，发现了用丛来研究Radon变换的有效性;（3）利用扭上同调群的交理论和外积，得到了广义超几何函数的一个对偶.在3）中，（1）给出了L-函数的Fourier系数积分表达式，并在一定条件下证明了L-函数的亚纯延拓和函数方程，它是Kohnen-Skoruppa关于二次Siegel尖形的结果的推广;（2）证明了局部域上具有正特征的p-adic Galoi表示范畴与具有Frobenius映射的标准微分模范畴，特别是惯性群因子的有限表示对应于过收敛模范畴之间的范畴等价，（3）研究了p-adic Hodge理论的层化及其相关性，它们类似于复流形上的Riemann-Hilbert对应。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