Abstract:
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.
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