On weakly pseudoconcave CR manifolds

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