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VERTEX OPERATOR ALGEBRAS AND MODULI SPACES OF ALGEBRAIC CURVES

顶点算子代数和代数曲线的模空间

基本信息

批准号：
15540036
负责人：
ICHIKAWA Takashi
金额：
$ 2.37万
依托单位：
SAGA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

项目摘要

(1)Using arithmetic Schottky-Mumford uniformization theory on algebraic curves, we constructed Teichmuller groupoids in the category of arithmetic geometry. By this construction, we gave a partial answer to Grothendieck's conjecture on the associated Galois representations, and described monodromy representations induced from confbrmal field theory.(2)Extending Ullmo-Zhang's results on Bogomolov's conjecture, we gave a condition that a subvariety of an abelian variety defined over a number field is isomorphic to an abelian variety in terms of Neron-Tate's height functions.(3)We described the structure of Riemann surfaces defined from the monodromy representation of hypergeometric differential equations with purely imaginary exponents (joint work with M.Yoshida).(4)We determined the structure of the class groups and unit groups of algebraic number fields of Kummer type, specifically of quartic Dirichlet fields (joint work with K.Katayama and C.Levesque). Further, we investigated the problem of Hasse concerning power integral basis of the ring of algebraic integers (joint work with Y.Motoda).(5)We defined stochastic holonomy operator and the Chern-Simons integral of some product of gauge invariant Wilson loop observables in the Wiener space setting.(6)We studied Buchsbaum Stanley-Reisner rings with linear resolution and characterized them by their multiplicity. Further, we studied the arithmetical rank and determined it for monomial ideals of deviation two.(7)We studied 4-manifolds having flexible surfaces inside, and showed that a lot of simply connected 4-manifolds not the 4-sphere have flexible surfaces. Further, we introduced an operation to alter any surface in a simply connected 4-manifold into a flexible surface.

(1)利用代数曲线上的算术肖特基-芒福德均匀化理论，构造了算术几何范畴内的Teichmuller群群。通过这种构造，我们对格洛腾迪克关于相关伽罗瓦表示的猜想给出了部分答案，并描述了由共模场论导出的一元表示。(2)将Ullmo-Zhang的结果推广到Bogomolov的猜想，我们给出了一个条件，即在数域上定义的阿贝尔簇的子簇与阿贝尔簇在Neron-Tate高度方面同构。 (3)我们描述了由纯虚指数超几何微分方程的单向表示定义的黎曼曲面的结构（与 M.Yoshida 合作）。(4)我们确定了 Kummer 型代数数域的类群和单位群的结构，特别是四次狄利克雷场（与 K.Katayama 和 C.Levesque 合作）。进一步，我们研究了关于代数整数环的幂积分基的 Hasse 问题（与 Y.Motoda 合作）。（5）我们定义了随机完整算子和维纳空间设置中规范不变 Wilson 环可观测量的某些乘积的 Chern-Simons 积分。（6）我们研究了具有线性分辨率的 Buchsbaum Stanley-Reisner 环，并用它们的特征来表征它们多重性。进一步研究了算术秩并确定了偏差二的单项式理想。(7)研究了内部具有柔性曲面的4-流形,结果表明许多单连通的4-流形而不是4-球面具有柔性曲面。此外，我们引入了一种将简单连接的 4 流形中的任何表面更改为柔性表面的操作。