Algebraic Varieties, Birational Geometry and the Structure of the Galois Groups

DMS-0404715 Fedor Bogomolov PI将继续研究光滑复射影簇的无限泛覆盖的几何和解析性质。这些都是相当复杂的分析变种，在大多数情况下都有无限的拓扑型。然而，在所有已知的情况下，这些簇都是全纯凸的，并且有结果证明了这一事实适用于广泛的射影簇。证明通常利用初始簇的基本群的表示的性质。其目的是为这些泛覆盖建立一个现实版本的全纯凸性猜想。尽管在许多重要情况下证明了泛覆盖的全凸性，但PI建议证明这种覆盖的存在包含无限紧曲线链，从而违反了全凸性。对于定义在有限域上的簇，PI焦点是建立定义在有限域上的亏格大于1的射影曲线的Torelli定理的类比，即每条这样的曲线都有一个嵌入到它的雅可比矩阵的点的扭群中.后者是一个无限扭群，它(几乎)只依赖于作为抽象群的初始曲线的亏格。因此，曲线的雅可比像提供了这个标准扭群中的一个无限子集，它由有限域的代数闭包上的曲线的所有点组成。其目的是证明该集合论映象将曲线完全定义为一个代数对象。本文的研究涉及代数几何、数论、群论和拓扑学。PI将研究定义在代数闭域上的代数簇几何的不同方面。对于定义在数域上的代数曲线，我们的研究将集中于找出一类极小曲线(猜想是一条曲线)，其性质是来自这类曲线的非分支覆盖支配于定义在代数数上的所有其他曲线。贝利定理指出，定义在数域上的代数簇的几何与定义在复数上的泛代数簇的几何有很大的不同。长期目标是为这一现象找到一个准确的公式。

DMS-0404715Fedor BogomolovThe PI is going to continue his study of the geometry and analytic properties of infinite universal coverings of smooth complex projective varieties. These are rather complicated analytic varieties, which in most of the cases have infinite topological type. However in all known cases these varieties are holomorphically convex and there are results establishing this fact for a broad range of projective varieties. The proofs usually exploit the properties of representations of the fundamental group of the initial variety. The objective is to establish a realistic version of holomorphic convexity conjecture for these universal coverings. Though holomoprhic convexity of the universal coverings was proved in many important cases the PI proposes to show the existence of such coverings containing infinite chains of compact curves thus violating holomoprhic convexity. For varieties defined over finite fields PI focus is to establish the analogue of Torelli theorem for the projective curves of genus greather than 1 defined over finite field.Namely every such curve has an imbedding into a torsion group of points of it's jacobian. The latter is an inifinite torsion group, which depends (almost) only on the genus of the initial curve as an abstract group. Thus the image of the curve in jacobian provides with an infinite subset of points in this standard torsion group consisting of all points of the curve over algebraic closure of the finite field. The objective is to show that this set theoretic image defines the curve completely as an algebraic object.The proposed research here lies at the interface of algebraic geometry, number theory, group theory and topology. The PI will study different aspects of the geometry of algebraic varieties defined over algebraically closed fields. For algebraic curves defined over number fields the research will focus on finding a minimal class of curves (conjecturally one curve) with the property that nonramified coverings of curves from the class dominate all the other curves defined over algebraic numbers. Bely's theorem indicated that the geometry of algebraic varieties defined over number fields substantially differs from the geometry of generic algebraic varieties over complex numbers. The long-term objective is to find a precise formulation for this phenomenon.