Chow Groups of Projective Varieties

Duke University, Durham NC

Investigators

Abstract

This project is concerned with the geometry of algebraic varieites. It builds on past experience which indicates that much can be learned about the geometry of a given algebraic variety by studying its Chow group. The Chow group is obtained from the free abelian group on all subvarieties of a given variety by modding out by the relation of rational equivalence. For algebraic curves the Chow group is a close relative of the Jacobian. While the Chow group is known to have good functorial properties, it is often very difficult to compute. Thus it is difficult to gain access to the subtle and valuable information that it frequently contains. This investigation focuses on three different aspects of the problem of computing Chow groups. First the torsion subgroup of the Chow group is being studied with an emphasis on the quotient by the subgroup generated by cycles algebraically equivalent to zero. Secondly, the relationship between the Chow group and singular cohomology with rational coefficients is being investigated. The Hodge Conjecture is an important concern here. The theory of Abelian varieties and techniques from classical algebraic geometry will be brought to bear on this problem. The third line of investigation involves the relationship between the Chow group and singular cohomology with integrer coefficients. An ancient guidepost here was the so called Integral Hodge Conjecture. It has turned out that the Integral Hodge Conjecture is false in some instances. The extent of its failure is poorly understood, so the principal investigator and a PhD student are working to clarify this. To a significant extent, the value of basic research in algebraic geometry results from the tendency of mathematicians to reduce problems in various other fields of mathematics to problems in algebraic geometry. This tendency is due to the ultimate simplicity of algebraic geometry, where remarkably effective approaches to certain geometric problems have been developed with minimal reliance on infinite processes. While the initial translation of a problem from the physical world into mathematics frequently involves limits, derivatives, integrals, and further notions involving infinite processes which go well beyond ordinary calculus, mathematicians have learned to search for hidden aspects of these problems which are essentially of an algebraic or finite nature. Neither the task of translating real world phenomena into mathematical problems, the task of discovering a hidden algebraic core in a problem formulated using infinite processes, nor the task of solving this core problem using algebraic geometry is often easy. Nonetheless, this lengthy process has led to profound insights. For example, algebraic geometry and some of the algebraic varieties studied by the principal investigator are of great interest to physicists in their current attempts to unify quantum mechanics with the theory of gravity.