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

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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.