Abstract
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.
Research supported by NSF G-24336.
Received May 1, 1964.
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References
H. Cartan Séminaire 1951–52, ENS Paris.
Serre, J.-P.: H. Cartan Séminaire 1951–52, ENS Paris, Exposé XX.
Serre, J.-P., H. Cartan Séminaire 1953–54, ENS Paris, Exposé XVIII.
Nickerson, H. K.: On the complex form of the Poincaré lemma. Proc. Amer. Math. Soc. 9, 182–188 (1958).
Hodge, W.V.D.: Differential forms on a Kähler manifold. Proc. Cambridge Phil. Soc. 47, 504–517 (1951).
Weil, A.: Introduction à l’étude des variétés kählériennes. Actualités scientifiques et industrielles 1267. Paris: Hermann 1958.
Aeppli, A.: Some exact sequences in cohomology theory for Kähler manifolds. Pac. J. Math. 12, 791–799 (1962).
Aeppli, A.: On determining sets in a Stein manifold. These Proceedings 48–58.
MacLane, S.: Homology. Springer — Academic Press 1963.
Stein, K.: Überlagerungen holomorph-vollständiger komplexer Räume. Arch. Math. VII 354–361, (1956).
Grauert, H.: Charakterisierung der holomorph-vollständigen komplexen Räume. Math. Ann. 129, 233–259 (1955).
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Aeppli, A. (1965). On the Cohomology Structure of Stein Manifolds. In: Aeppli, A., Calabi, E., Röhrl, H. (eds) Proceedings of the Conference on Complex Analysis. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-48016-4_7
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