Cycles, characters and global geometry

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

Abstract Award: DMS-0404766 Principal Investigator: H. Blaine Lawson, Jr. This project is concerned with the study of cycles, residues, boundaries and differential characters. The proposal has several interrelated parts. The first concerns the groups of algebraic cycles and cocycles on a projective variety $X$. The aim is to relate these groups to the global structure of $X$. The investigator has, with others, established a theory of homology type for algebraic varieties based on the homotopy groups of cycles spaces. This theory will be used to study concrete questions about algebraic spaces. Implications for real algebraic geometry will be explored. Striking onnections to universal constructions in topology which emerged in prior research will also be investigated. A second part of the proposal concerns cycles which bound complex subvarieties in a projective manifold. Several new conjectures relate these cycles to approximation theory, pluripotential theory, and the projective spectrum of Banach graded algebras. A third area of the proposal concerns the study of singularities and characteristic forms. This subject includes a generalization of Chern-Weil theory which gives canonical homologies between singularities of bundle maps and characteristic forms. It includes a useful analytic tool -- geometric atomicity -- which will be studied, and it yields a new approach to Morse Theory. Applications relating singularities to global geometry remain to be investigated. The forth part of the proposal concerns sparks and spark complexes. This recently developed framework for the study of differential characters has yielded interesting generalizations which extend Deligne cohomology and arithmetic Chow groups. They are essentially secondary invariants which mediate between cycles and smooth data. Further development of the theory and its application to the study of cycles is proposed. A fifth area is concerned with special cycles in geometry, in particular Special Lagrangian cycles in Calabi-Yau manifolds, and associative and Cayley cycles in $G_2$ and Spin$_7$ spaces. These latter subjects relate to mirror symmetry conjectures and to M-theory in Physics as well as many areas of geometry and algebra. This project will also be concerned with student development, including an undergraduate educational effort aimed at fostering mathematical independence and developing interactive enviornments. A concept of central importance in geometry is that of a ``cycle''. In algebraic geometry a cycle corresponds to the simultaneous solution of a system of polynomial equations. In differential geometry they arise in many ways: as the large scale solutions of certain differential equations, and as the level sets and singularity sets of differentiable mappings. Curves and surfaces in space are simple examples. Cycles with a particular geometry also play a fundamental role in modern physical theories This proposal is concerned with the study of cycles across this broad spectrum. In the algebraic setting cycles have been related to fundamental large-scale geometry of their surrounding space. This discovery has revealed surprizing and important relationships between spaces of algebraic cycles and fundamental constructions in algebraic topology and has led to new insights in both fields. This work will be continued. Another area of investigation concerns cycles which form the boundary of subsets with special geometric structure. They represent non-linear versions of classical boundary value problems in analysis. Such questions arise in many contexts. Recently the proposer has formulated conjectures relating certain important classes of such cycles to questions in approximation theory and Banach algebras. Successful resolution should produce significant new insights in several fields of mathematics. A third area of study concerns a mathematical apparatus developed by the proposer to detect subtle relationships between cycles and the global structure of the space they live in. This apparatus encompasses some of the most effective tools historically developed for this purpose, and it is much more general. Further development of this theory and its applications will be persued. A fourth domain of investigation is concerned with special cycles in geometry: Special Lagrangian cycles in Calabi-Yau manifolds, and associative and Cayley cycles in G(2) and Spin(7) spaces. These latter subjects relate to gauge field theory and gravity in Physics This project will also be concerned with graduate student development. Students will be part of the research team. There will also be an undergraduate educational effort aimed at fostering mathematical independence and developing interactive environments.