A Dolbeault isomorphism theorem in infinite dimensions

Simon, Scott

doi:10.1090/S0002-9947-08-04550-9

A Dolbeault isomorphism theorem in infinite dimensions
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Trans. Amer. Math. Soc. 361 (2009), 87-101 Request permission

Abstract:

For a large class of separable Banach spaces, we prove the real analytic Dolbeault isomorphism theorem for open subsets.

References

Henri Cartan, Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. France 85 (1957), 77–99 (French). MR 94830
Pierre Dolbeault, Sur la cohomologie des variétés analytiques complexes, C. R. Acad. Sci. Paris 236 (1953), 175–177 (French). MR 52771
Hans Grauert, On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460–472. MR 98847, DOI 10.2307/1970257
László Lempert, The Dolbeault complex in infinite dimensions. I, J. Amer. Math. Soc. 11 (1998), no. 3, 485–520. MR 1603858, DOI 10.1090/S0894-0347-98-00266-5
László Lempert, Approximation of holomorphic functions of infinitely many variables. II, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 2, 423–442 (English, with English and French summaries). MR 1775356
László Lempert, The Dolbeault complex in infinite dimensions. III. Sheaf cohomology in Banach spaces, Invent. Math. 142 (2000), no. 3, 579–603. MR 1804162, DOI 10.1007/PL00005794
László Lempert, Plurisubharmonic domination, J. Amer. Math. Soc. 17 (2004), no. 2, 361–372. MR 2051614, DOI 10.1090/S0894-0347-03-00448-X
László Lempert, Vanishing cohomology for holomorphic vector bundles in a Banach setting, Asian J. Math. 8 (2004), no. 1, 65–85. MR 2128298, DOI 10.4310/AJM.2004.v8.n1.a7
Jorge Mujica, Complex analysis in Banach spaces, North-Holland Mathematics Studies, vol. 120, North-Holland Publishing Co., Amsterdam, 1986. Holomorphic functions and domains of holomorphy in finite and infinite dimensions; Notas de Matemática [Mathematical Notes], 107. MR 842435
Imre Patyi, On the $\overline \partial$-equation in a Banach space, Bull. Soc. Math. France 128 (2000), no. 3, 391–406 (English, with English and French summaries). MR 1792475
Ivan Singer, Bases in Banach spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 154, Springer-Verlag, New York-Berlin, 1970. MR 0298399
R. O. Wells Jr., Differential analysis on complex manifolds, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0515872