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

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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.