Singularities and Chern-weil Theory, Ii: Geometric Atomicity

SINGULARITIES AND CHERN-WEIL THEORY, II Geometric Atomicity

We study the relationship between characterstic classes and singularities for a general smooth bundle map : E ! F . We show that there exist explicit and canonical equations of the form ( ) ( ) = k Resk [ k( )] + dT for every characteristic form ( ) of either E or F , provided that satis es the condition of geometric atomicity. This condition, which is a central concept of the paper, is quite b...

Lectures on Algebro-geometric Chern-weil and Cheeger-chern-simons Theory for Vector Bundles

Chern-weil Theory and Some Results on Classic Genera

In this survey article, we first review the Chern-Weil theory of characteristic classes of vector bundles over smooth manifolds. Then some well-known characteristic classes appearing in many places in geometry and topology as well as some interesting results on them related to elliptic genera which involve some of my joint work in [HZ1, 2], are briefly introduced. 1. Chern-Weil theory for chara...

Geometric interpretation of simplicial formulas for the Chern-Simons invariant

Chern-Simons theory was first introduced by S. S. Chern and J. Simons in [CS] as secondary characteristic classes: given a Lie group G and a flat Gbundle P over a manifoldM , all Chern-Weil characteristic classes of P has to vanish as P have a flat connection whereas the bundle might be non trivial. The Chern-Simons functional is a non trivial invariant of G-bundles with connections. This theor...

Differential cohomology in a cohesive ∞-topos

We formulate differential cohomology and Chern-Weil theory the theory of connections on fiber bundles and of gauge fields abstractly in the context of a certain class of higher toposes that we call cohesive. Cocycles in this differential cohomology classify higher principal bundles equipped with cohesive structure (topological, smooth, synthetic differential, supergeometric, etc.) and equipped ...