Vol. 39, No. 2, 1971

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Cohomology groups associated with the operator

Bohumil Cenkl and Giuliano Sorani

Vol. 39 (1971), No. 2, 351–369
Abstract

Let M be a complex analytic manifold of complex dimension m. The manifold M, considered open, is a submanifold of a manifold Mof the same dimension, and its boundary M is a smooth C8-manifold. Let Ap,q be the sheaf of germs of complex-valued (p,q)-forms, p and q are integers, p 0,q 0. The exterior differential of an element u Ap.q can be written in a unique way as a sum du = ∂u + u. There is a real operator

deu = √ − 1(∂u − ∂u)

and the real second order operator

       √--- --
dde = 2 − 1∂∂

defined on Apq. Let ARp,q = {α = α1 + α2 Ap.q Aq,p|α2 = α1} be the sheaf of real (p,q)-forms. Then we get two short exact sequences of sheaves

                  ∂∂          d
0 −→ 𝒫q,Cq− → Ap,q−→  Ap+1,q+1 −→ Ap+2,q+1 ⊕ Ap+1,q+2
p,q      p,q ddc  p+1,q+1  d   p+2,q+1
0 −→ 𝒫 R − → A R −→  AR      −→ A R     ⊕ Ap+1,q+2
(1.1)

where 𝒫cp,q and 𝒫Rp,q are defined by these sequences. The purpose of this paper is to discuss the cohomology of these two sequences.

Mathematical Subject Classification 2000
Primary: 32C35
Secondary: 58G05
Milestones
Received: 3 August 1970
Published: 1 November 1971
Authors
Bohumil Cenkl
Giuliano Sorani