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On the $$\bar\partial$$-equation in a Banach space. (English. English, French summaries) Zbl 0967.32036
Summary: We define a separable Banach space $$X$$ and prove the existence of a $$\overline \partial$$-closed {$$C^\infty$$}-smooth $$(0,1)-f$$ on the unit ball $$B$$ of $$X$$, which is not $$\overline\partial$$-exact on any open subset. Further, we show that the sheaf cohomology groups $$H^q(\Omega, {\mathcal 0})=0$$, $$q\geq 1$$, where $${\mathcal 0}$$ is the sheaf of germs of holomorphic functions on $$X$$, and $$\Omega$$ is any pseudoconvex domain in $$X$$, e.g., $$\Omega=B$$. As the Dolbeault group $$H^{0,1}_{ \overline\partial}(B)\not=0$$, the Dolbeault isomorphism theorem does not generalize to arbitrary Banach spaces. Lastly, we construct a {$$C^\infty$$}-smooth integrable almost complex structure on $$M=B\times\mathbb{C}$$ such that no open subset of $$M$$ is biholomorphic to an open subset of a Banach space. Hence the Newlander–Nirenberg theorem does not generalize to arbitrary Banach manifolds.

##### MSC:
 32W05 $$\overline\partial$$ and $$\overline\partial$$-Neumann operators 32L20 Vanishing theorems 58B12 Questions of holomorphy and infinite-dimensional manifolds 32Q99 Complex manifolds 46G20 Infinite-dimensional holomorphy
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