Algebraic varieties, birational geometry and the structure of the Galois groups

首席研究员计划研究代数几何、代数和数论中的几个问题。其中一个目标是了解一个小余维的复射影空间的光滑子簇的切线几何与这种子簇上的全纯张量结构之间的关系。这将提供满足特殊切条件的子簇的描述。在另一个项目的计划是要表明，以前构建的例子投影表面确实提供了反例的一致化猜想，声称全纯凸的普遍覆盖的复杂的投影流形。该项目的完成将改变目前对非单连通射影簇结构的看法。另一个项目的主要目标是一个优势猜想的特殊类别的双曲曲线定义的代数数。支配猜想认为，任何这样的曲线都有一个有限的非分歧覆盖，覆盖到任意其他曲线上。覆盖取决于目标曲线。这一结果将提供一个简单的几何原因代数点的结构上的所有算术双曲曲线是相似的。因此，关于双曲曲线的算术的许多陈述将足以仅检查一条这样的曲线。它为代数簇算术的发展开辟了一条新的途径。另一个项目的重点是研究定义在有限域上的双曲曲线的结构，该有限域被认为是在标准嵌入到曲线的雅可比矩阵中的点的子集。在这种情况下，雅可比矩阵被简单地看作是一个无限挠群。以这种方式定义的子集有许多显着的属性，计划是系统地研究这些属性。特别是其中一个目标是检查是否上述子集定义完全领域的代数函数的曲线。 复代数簇几何的研究与数学的其他领域和现代物理学的几个领域有许多联系。射影空间中的小余维光滑子流形具有非常特殊的几何性质，对它们的研究是由一个猜想激发的，这个猜想预言了对这种子流形的一个简单而完整的描述。小域上的代数曲线现在出现在许多应用中。与复代数曲线相比，这些曲线具有许多额外的几何性质，目标是揭示和描述它们。

The principal investigator plans to work on several problems in algebraic geometry, algebra and number theory. One of the objectives is to understand a relation between geometry of tangencies of smooth subvarieties of a complex projective space of small codimension with a structure of holomorphic tensors on such subvarieties. This will provide a description of such subvarieties satisfying special tangent conditions. In another project the plan is to show that previously constructed examples of projective surfaces indeed provide counter examples to the uniformization conjecture which claims holomorphic convexity of the universal coverings of complex projective manifolds. The completion of this project will change the current perception of the structure of non-simply-connected projective varieties. The main objective of another project is a dominance conjecture for special classes of hyperbolic curves defined over algebraic numbers. The dominance conjecture claims that any such curve has a finite nonramified covering which surjects onto arbitrary other curve. The covering depends on the target curve. This result would provide a simple geometric reason for the structure of algebraic points on all arithmetic hyperbolic curves to be similar. Hence many statements about the arithmetic of hyperbolic curves would be sufficient to check for one such curve only. It opens a new avenue for the advance in arithmetics of algebraic varieties. The focus of another project is a study of the structure of a hyperbolic curve defined over a finite field considered as a subset of its points under a standard imbedding into the jacobian of the curve. In this context the jacobian is viewed simply as an infinite torsion group. The subsets defined in this way have many remarkable properties and the plan is to investigate those systematically. In particular one of the objectives is to check whether the above subset defines completely the field of algebraic functions on the curve. The study of the geometry of complex algebraic varieties has many connections to other areas of mathematics and several areas of modern physics. Smooth submanifolds of small codimension in a projective space have very special geometry and their study is motivated by a conjecture which predicts a simple and complete description of such manifolds. Algebraic curves over small fields appear now in a multitude of applications. As it happens these curves have many additional geometric properties compare to complex algebraic curves and the objective is to uncover and describe them.