| The purpose of this dissertation is to study secondary geometric invariants of smooth manifolds like Cheeger-Simons differential characters, Deligne cohomology and Harvey-Lawson spark characters. Our approach follows Harvey-Lawson spark theory. In particular, we study these secondary geometric invariants via the presentation of smooth hypersparks and give a new description of the ring structure of differential characters. We also study d¯-spark theory and the ring functor H*(·, p) of complex manifolds which is a natural extension of Deligne cohomology. We represent Deligne cohomology classes by d¯-sparks and give an explicit product formula for Deligne classes. Massey higher products of secondary geometric invariants are also studied. Moreover, we show a Chern-Weil-type construction of Chern classes in Deligne cohomology for holomorphic vector bundles over complex manifolds. Many applications of our theory are given. Generalized Nadel invariants are defined naturally from our construction of Chern classes and Nadel's conjecture is verified. Studying Chern classes for the normal bundles of holomorphic foliations, we establish an analogue of the Bott vanishing theorem. Applying our representation of analytic Deligne cohomology classes, we give a direct proof of the well known cycle map psi : CH*(X) → H2*DX,Z * . |
