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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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References:
[1] COEURÉ (G.) . - Les équations de Cauchy-Riemann sur un espace de Hilbert , manuscript.
[2] DINEEN (S.) . - Cousin’s first problem on certain locally convex topological vector spaces , Ann. Acad. Brasil. Cienc., t. 48, 1976 , p. 11-12. MR 55 #11290 | Zbl 0354.46029 · Zbl 0354.46029
[3] HÖRMANDER (L.) . - An Introduction to Complex Analysis in Several Variables , 3rd edition. - North-Holland, Amsterdam, 1990 . Zbl 0685.32001 · Zbl 0685.32001
[4] JACOBOWITZ (H.) . - Simple examples of nonrealizable CR hypersurfaces , Proc. Amer. Math. Soc., t. 98, 1986 , p. 467-468. MR 87k:32034 | Zbl 0605.32009 · Zbl 0605.32009
[5] LEBRUN (C.) . - A Kähler structure on the space of string world sheets , Class. Quant. Gravity, t. 10, 1993 , p. 141-148. MR 94m:58009 | Zbl 0790.53052 · Zbl 0790.53052
[6] LEMPERT (L.) . - The Dolbeault complex in infinite dimensions I , J. Amer. Math. Soc., t. 11, 1998 , p. 485-520. arXiv | MR 99f:58007 | Zbl 0904.32014 · Zbl 0904.32014
[7] LEMPERT (L.) . - The Dolbeault complex in infinite dimensions II , J. Amer. Math. Soc., t. 12, 1999 , p. 775-793. arXiv | MR 2000e:32053 | Zbl 0926.32048 · Zbl 0926.32048
[8] LEMPERT (L.) . - The Dolbeault complex in infinite dimensions III , to appear in Invent. Math. arXiv | Zbl 0983.32010 · Zbl 0983.32010
[9] LEMPERT (L.) . - Approximation de fonctions holomorphes d’un nombre infini de variables , Ann. Inst. Fourier, Grenoble, t. 49, 1999 , p. 1293-1304. Numdam | MR 2001d:32027 | Zbl 0944.46046 · Zbl 0944.46046
[10] LEMPERT (L.) . - Loop spaces as complex manifolds , J. Diff. Geom., t. 38, 1993 , p. 519-543. MR 94m:58010 | Zbl 0792.58008 · Zbl 0792.58008
[11] LEMPERT (L.) . - Approximation of holomorphic functions of infinitely many variables II , to appear in Ann. Inst. Fourier, Grenoble. Numdam | Zbl 0969.46032 · Zbl 0969.46032
[12] MAZET (P.) . - Analytic Sets in Locally Convex Spaces . - North-Holland Math. Studies 89, Amsterdam, 1984 . MR 86i:32012 | Zbl 0588.46032 · Zbl 0588.46032
[13] MEISE (R.) , VOGT (D.) . - Counterexamples in holomorphic functions on nuclear Fréchet spaces , Math. Z., t. 182, 1983 , p. 167-177. Article | MR 84m:46048 | Zbl 0509.46041 · Zbl 0509.46041
[14] NEWLANDER (A.) , NIRENBERG (L.) . - Complex analytic coordinates in almost complex manifolds , Ann. of Math., t. 65, 1957 , p. 391-404. MR 19,577a | Zbl 0079.16102 · Zbl 0079.16102
[15] RYAN (R.A.) . - Holomorphic mappings in l1 , Trans. Amer. Math. Soc., t. 302, 1987 , p. 797-811. MR 88h:46089 | Zbl 0637.46045 · Zbl 0637.46045
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