The Serre duality theorem for a non-compact weighted CR manifold

Abstract

Abstract. It is proved that the Hodge decomposition and Serre duality hold on a non-compact weighted CR manifold with negligible boundary. A complete CR manifold has negligible boundary. Some examples of complete CR manifolds are presented.

Vanishing and conservativeness of harmonic forms of a non-compact CR manifold

“…Finally, let us point out that our results extend to any holomorphic vector bundle valued forms over a weighted CR manifold (see e.g. [15]).…”

“…in the last passage we have used the fact that T (e i , e j ) = δ ij √ −1ξ . Therefore, by the Ricci identity, (15)…”

“…Therefore, by combining Lemmae 13 and 5, if M is Riemannian complete, it has negligible boundary. All of the following examples are Riemannian complete (see [15] for details). EXAMPLE 1 (Heisenberg group).…”

“…ACKNOWLEDGMENTS. The author wishes to express his gratitude to Dr. Itoh and Mr. Saotome for providing him with interesting examples of non-compact CR manifolds that are studied from a different point of view in the joint work [15].…”