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Zbl 0967.32036
Patyi, Imre
On the $\bar\partial$-equation in a Banach space. (English. English, French summaries)
[J] Bull. Soc. Math. Fr. 128, No.3, 391-406 (2000). ISSN 0037-9484

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 $H^q(\Omega, {\Cal 0})=0$, $q\ge 1$, where ${\Cal 0}$ 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 $M=B\times\bbfC$ 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 2000:: *32W05 $\overline\partial$ and $\overline\partial$-Neumann operators
32L20 Vanishing theorems (analytic spaces)
58B12 Questions of holomorphy in infinite-dimensional manifolds
32Q99 Complex manifolds
46G20 Infinite dimensional holomorphy

Keywords: $\overline\partial$ equation; Dolbeault isomorphism; Newlander-Nirenberg theorem; separable Banach space

Cited in: Zbl 1225.32042

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