On Weakly Pseudoconcave CR Manifolds

Abstract

In [Ann. Mat. Pura Appl. (4) 182 (2003), no. 1, 103--112; MR1970466 (2004k:32056)], C D. Hill and author extended idea of pseudoconcavity of a CR manifold to that of weakly pseudoconcave CR manifolds. In that paper and in [Rend. Sem. Mat. Univ. Padova 111 (2004), 179--204; MR2076739 (2005j:32039)], Hill and author studied properties of CR functions and CR meromorphic functions on these weakly pseudoconcave CR manifolds. As abstract states, this paper investigates the consequences of this weaker assumption of pseudoconcavity, together with finite kind, for top degree cohomology of tangential Cauchy-Riemann complex''. Section one is introduction. Section two gives basic needed definitions, including that of weakly pseudoconcave. As is to be expected, this definition captures requirement that Levi form has certain positive and negative eigenvalues. Section three is heart of paper, stating and proving various finiteness and vanishing theorems. The final section gives three examples. {For entire collection see MR2298775 (2007j:35005).}