Rationality of renormalized Chern classes

Abstract

Renormalized Chern classes (c) over tilde (2),...,(c) over tilde (n) for a compact, connected, complex manifold X with smooth, strictly pseudoconvex boundary M were defined in [7]. A Chern-Simons type invariant was also defined in [6] for a compact, strictly pseudoconvex CR-manifold M of dimension three, and a Gauss-Bonnet theorem relating the two in [7]. We show here that if M is spherical, and P((c) over tilde (2),...(c) over tilde (n)) is a polynomial with rational coefficients in (c) over tilde (2),...(c) over tilde (n), then the corresponding Chern class P(X) = P((c) over tilde (2),...,(c) over tilde (n)) defines a class in H* (X, M; Q). The proof is based on the multi-valued Hartogs theorem of [5] and the exponents of monodromy for a development map.