Geodesic
On a~Stein manifold the Dolbeault complex splits in positive dimensions
Sbornik. Mathematics, Tome 17 (1972) no. 2, pp. 289-316

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In this paper we find necessary and sufficient conditions for the $\overline\partial$ operator, acting in the Dolbeault complex of an analytic locally free sheaf of finite type on a complex manifold, to split in a given dimension, i.e. to possess a linear continuous right inverse operator. In particular, from this it follows that on a Stein manifold the $\overline\partial$ operator always splits in all positive dimensions, while it does not split in dimension zero. We also consider some questions connected with this; in particular, the splitting of operators in the Frechet spaces and the splitting of the de Rham complex on a differentiable manifold. Bibliography: 11 titles. 
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     author = {V. P. Palamodov},
     title = {On {a~Stein} manifold the {Dolbeault} complex splits in positive dimensions},
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V. P. Palamodov. On a~Stein manifold the Dolbeault complex splits in positive dimensions. Sbornik. Mathematics, Tome 17 (1972) no. 2, pp. 289-316. http://geodesic.mathdoc.fr/item/SM_1972_17_2_a9/