9615808 Gajer This project lies on the interface of algebraic topology, differential geometry, and algebraic geometry. It is based on the investigator's recent work, in which he successfully employed some classical constructions of algebraic topology and differential geometry to reveal the geometric content of ordinary cohomology as well as smooth and holomorphic Deligne cohomology. The investigator plans to apply the ideas contained in his previous research on Deligne cohomology to study the structure of the image and the kernel of the cycle homomorphism. Some of the deepest and most important problems of transcendental algebraic geometry concern the cycle homomorphism. The most celebrated among these is the Hodge conjecture. Except for a few classes of very special algebraic varieties, codimension one is the only codimension in which complete cohomological and geometric descriptions of the image and the kernel of the cycle homomorphism exist. The investigator intends to pursue extensions to all codimensions of structures playing a central role in the codimension one case. In particular, he introduces generalized exponential sequences and a new cohomology theory that is conjecturally dual to Chow groups of algebraic cycles modulo rational equivalence. The objective of this project is to verify that conjecture. If this is the case, then the cohomology long exact sequence of the generalized exponential sequence will induce a complete cohomological description of the image and the kernel of the cycle homomorphism. Projective varieties are geometric objects defined by means of polynomial equations. Like other notions that arise in many important contexts and sound deceptively simple, for example, the ordinary whole numbers, they possess properties that are not at all obvious or easy to discover, and over the course of many years, mathematicians have been led to develop elaborate algebraic machinery for making computations concerning some of these properties. It is hard to remain in touch with the geometric meaning of these computations, and this is what has given rise to the current project. The investigator will attempt to understand a geometric structure of what are known as algebraic cycles. This will be done by extending some classical techniques of algebraic topology and differential geometry to the context of algebraic geometry. One might say that at least this one corner of algebraic geometry will be reintroduced to its roots. This will be not only an esthetic triumph, but an aid to intuition and thus to further discoveries and advances in mathematics and in the disciplines like theoretical physics that make use of this mathematics. ***
