Complex cobordism and embeddability of CR-manifolds

This paper studies complex cobordisms between compact, three dimensional, strictly pseudoconvex Cauchy-Riemann manifolds. Suppose the complex cobordism is given by a complex 2-manifold X with one pseudoconvex and one pseudoconcave end. We answer the following questions. To what extent is X determined by the pseudoconvex end? What is the relation between the embeddability of the pseudoconvex end and the embeddability of the pseudoconcave end of X? Do all CR-functions on the pseudoconvex end of X extend to holomorphic functions on the interior of X? We introduce two methods to construct pseudoconcave surfaces that show that the complex 2-manifold X giving a complex cobordism is not determined by the pseudoconvex end. These two constructions give new methods to construct non-embeddable Cauchy-Riemann 3-manifolds and prove that embeddability of a strictly pseudoconvex Cauchy-Riemann 3manifold is not a complex-cobordism invariant. We show that a new phenomenon occurs: there are CR-functions on the pseudoconvex end that do not extend to holomorphic functions on X. We also show that the extendability of the CR-functions from the pseudoconvex end is necessary but not sufficient for embeddability to be preserved under complex cobordisms. A compact (2n+1)-dimensional Cauchy-Riemann manifold (CR-(2n+1)-manifold) consists of: a compact (2n+1)-dimensional manifold M , a rank n complex subbundle T M ⊂ TM ⊗ C satisfying T M ∩ T M = {0} (T M ≡ T 0,1M) and the integrability condition [Z,Z ′] ∈ T M for local sections Z,Z ′ ∈ T M . If the CR-(2n+1)-manifold has the additional property that any nonvanishing local section Z of T M is such that [Z,Z] 6∈ T M ⊕ T M , it is called strictly pseudoconvex (SPCR-(2n+1)-manifold). A differentiable function f : M → C is said to be a CR function if it verifies the tangential Cauchy-Riemann equations Zf = 0 for all local sections of T M . In other words, CR functions are the elements of the kernel of the ∂b operator defined as ∂bf = df |T 0,1M .