On the Cohomology Structure of Stein Manifolds

W. V. D. Hodge considered in [5] differential forms on a compact Kahler manifold and proved certain “natural” isomorphisms between mixed cohomology groups and modules of harmonic forms. Here we study these mixed groups on a Stein manifold X (for the definition and properties of Stein manifolds see [1], [11]) and get the isomorphisms $$ H_{d/\nabla }^{p,q} (X) \cong H^{p + q} (X;C),H_{\nabla /d',d''}^{p,q} (X) \cong H^{P + q + 1} (X;C)for\:p,q \geqq 1 $$ and $$ H_{(d/d')a''}^{p,q} (X) \cong H^{p + s} (X;C)for\;p \geqq 1 $$ (Theorems 1 and 2 in Section 4). This last isomorphism generalizes Serre’s isomorphism given in [2]. We discuss naturality in Section 5: the mentioned isomorphisms are induced by the obvious imbeddings of forms and with the help of an isomorphism $$ d':H_{\nabla /d',d''}^{p,q} (X) \to H_{(d/d')d''}^{p + 1,q} (X)(p,q \geqq 1) $$ . As a result we state in Corollaries 1 and 2: every d-exact (p + q)-form, p + q ≧ 1, on X is d-cohomologous to a pure type (p, q)-form, and a d-total (p, q)-form, p, q ≧ 1, on X is ∇-total. In Section 6 the relative d/∇ cohomology groups are treated: Theorem 4 asserts $$ H_{d/\nabla }^{p,q} (K,L) \cong H^{p,q} (K,L;C)if\;p,q \geqq 2 $$ for a pair of Stein manifolds, and in Theorem 5 a short exact sequence is given relating mixed groups with relative mixed groups in case of a pair (X, ∂X̃) where ∂X̃ is a suitable open neighborhood (in X) of the boundary of X.