This project is concerned with the study of cycles and their boundaries, forms, and generalized plurisubharmonic functions. 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 connections to universal constructions in topology which emerged in prior research will also be investigated. The second part of the proposal concerns cycles which bound holomorphic chains in projective manifolds. In particular, characterizations in terms of projective linking numbers and quasi-plurisubharmonic functions will be sought. This will entail a deep analysis of the structure of projective hulls, a concept analogous to polynomial hulls, which has been introduced by the investigator and is ofindependent interest. Projective hulls are related to approximation theory, pluripotential theory, and the spectrum of Banach graded algebras. The third topic, a major part of the proposal, concerns the broad development of a pluripotential theory in calibrated and other geometries. The notions of plurisubharmonic function, pseudo-convex domain, capacity, and solutions to the Dirichlet problem for generalized Monge-Amp`ere-type equations will be studied in a very general setting. This project, already underway, should have an impact in calibrated geometry, which in turn plays an important role in M-theory in modern physics. There should also be applications to symplectic geometry and to p-convexity in riemannian geometry. The forth part of the proposal concerns sparks and spark complexes. These objects mediate betweeen cycles and smooth data, and give a concrete presentation of differential characters and their generalizations. In the complex category this involves an analytic study of Deligne cohomology and relates to arithmetic Chow groups. It yields invariants for bundles and foliations, and retrieves the classical Abel-Jacobi mappings. This project will also be concerned with student development, including an undergraduate educational effort aimed at fostering mathematical independence and developing interactive environments.
Part II
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. The proposer has formulated conjectures relating important classes of such cycles to questions in approximation theory and Banach algebras. Successful resolution will establish a series of new results in complex geometry and should lead to significant new insights in several other fields of mathematics.
A third part of the proposal aims at extending classical pluripotential theory to very general geometric settings. These include calibrated geometries, symplectic and Lagrangian geometries, and much more.An uncanny amount of the classical theory has already been shown to hold in this general context. Solutions to the Dirichlet problem for associated Monge-Ampere type equations will be sought in this setting. The study is, in a certain strict sense, dual to the study of the special cycles appearing in these geometries. It should apply to 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
A forth domain of investigation concerns a mathematical apparatus developed by the proposer and R. Harvey 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 pursued.
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.
