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

fedor bogomolov PI将继续他对光滑复射影变数的无穷泛覆盖的几何和解析性质的研究。这些是相当复杂的解析变异体，在大多数情况下具有无穷拓扑型。然而，在所有已知的情况下，这些变种是全纯凸的，有结果建立了这一事实，为广泛的射影变种。这些证明通常利用初始变量的基本群的表示的性质。目的是为这些全纯覆盖建立一个真实版本的全纯凸性猜想。虽然在许多重要的情况下证明了泛覆盖的全全凸性，但PI提出了证明这种包含紧曲线的无限链从而违反全全凸性的覆盖的存在性。对于限定在有限域上的变量，重点是建立限定在有限域上的大于1的属的射影曲线的Torelli定理的类似。也就是说，每一个这样的曲线都有一个嵌入到它的雅可比矩阵的扭转组中。后者是一个无限扭转群，它（几乎）只依赖于作为抽象群的初始曲线的属。因此，曲线在雅可比矩阵中的像提供了由有限域的代数闭包上曲线的所有点组成的标准扭转群中的点的无限子集。目的是证明这个集合论图像将曲线完全定义为一个代数对象。本文的研究方向是代数几何、数论、群论和拓扑学的交叉领域。PI将研究在代数闭域上定义的代数变量的几何的不同方面。对于定义在数域上的代数曲线，研究将集中于寻找最小曲线类（推测为一条曲线），其性质是该类曲线的非分支覆盖优于所有其他定义在代数数上的曲线。别利定理指出在数域上定义的代数变量的几何与在复数上定义的一般代数变量的几何有本质的不同。长期目标是为这一现象找到一个精确的表述。

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.