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。其中一个目的是了解小余维复射影空间的光滑子簇的切几何与这些子簇上的全纯张量结构之间的关系。这将提供满足特殊切线条件的这些亚种的描述。在另一个项目中，计划证明先前构造的射影曲面的例子确实提供了与齐化猜想相反的例子，该猜想要求复射影流形的泛覆盖的全纯凸性。该项目的建成将改变目前人们对非单连通投射品种结构的看法。另一个项目的主要目标是关于定义在代数数上的特殊类型的双曲曲线的优势猜想。支配猜想声称，任何这样的曲线都有一个有限的非分支覆盖，它射影到任意其他曲线上。覆盖取决于目标曲线。这一结果将为所有算术双曲曲线上的代数点的结构相似提供一个简单的几何原因。因此，许多关于双曲线算术的陈述将足以仅检查一条这样的曲线。它为代数簇的算术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下的点的子集。在这种情况下，雅可比被简单地视为一个无限扭群。以这种方式定义的子集具有许多显著的性质，我们的计划是系统地研究这些性质。特别地，其中一个目的是检查上述子集是否完全定义了曲线上的代数函数域。研究复代数簇的几何与数学的其他领域和现代物理学的几个领域有许多联系。射影空间中余维较小的光滑子流形具有非常特殊的几何性质，它们的研究是由一个猜想推动的，这个猜想预言了这种流形的简单而完整的描述。现在，小域上的代数曲线出现在许多应用中。碰巧的是，与复杂的代数曲线相比，这些曲线具有许多额外的几何性质，其目的是揭示和描述它们。

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.