Chow Groups of Smooth Projective Varieties

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This research project focuses on algebraic varieties, the geometric objects defined by the solution sets of systems of polynomial equations. Varieties will be studied through their Chow groups. The Chow group of a variety is the free abelian group on subvarieties modulo the subgroup of elements which are rationally equivalent to zero. Although the Chow group is known to have good abstract, functorial properties little is known about its structure for most varieties. The goal of the project is to make progress towards a better understanding of various aspects of this structure. The torsion in the Chow group will be studied as will the relationship between the Chow group and various cohomology groups. Furthermore, the quotient of the Chow group by the relation of algebraic equivalence will be investigated. Special properties which occur when the variety is defined over a finite field will also be studied.The line of investigation pursued in this project continues historical investigations with roots in the 19-th century. The research should contribute to a better understanding of the geometric figures defined by the solution sets to systems of polynomial equations. It is also motivated by a large body of conjectures, some recent and some quite old, which predict a beautiful and subtle relationship between geometry and the theory of numbers.