Secondary Characteristic Classes in CR Geometry

Abstract

We construct a family of $\mathbb{R}/\mathbb{Z}$-valued CR invariants for compact strictly pseudoconvex CR manifolds admitting pseudo-Einstein contact forms. We define these invariants by applying the theory of Cheeger--Simons differential characters to a globally defined modification of the normal tractor connection. When the CR holomorphic tangent bundle is trivial, their natural $\mathbb{R}$-valued lifts agree with the generalized Burns--Epstein invariants. For CR manifolds arising as boundaries of relatively compact strictly pseudoconvex domains, we identify these differential characters with those determined by the renormalized connection and derive bulk--boundary formulas involving renormalized characteristic numbers and residues of Chern classes. These formulas recover and extend the results of Burns--Epstein and Marugame. We also obtain obstructions to CR embeddings into complex Euclidean space.