On the Levi problem in Riemann domains over infinite dimensional Grassmann manifolds Masaru Nishihara Fukuoka Institute of Technology Oka [6], [7] proved that a pseudoconvex Riemann domain over C n is a domain of holomorphy, that is, he solved affirmatively the Levi problem in a Riemann domain over C n . The result of Oka has been extended to Riemann domains over a complex manifold(cf. Siu [8] and its references therein). Moreover his result has been extended to Riemann domains over various infinite dimensional topological vector spaces(cf. Dineen [1], [2] and their references therein). Let E be a complex Banach space with a Schauder basis and let G(E; r) be the Grassmann manifold of all r-dimensional complex linear subspaces in E. Let (ω, ϕ) be a Riemann domain over G(E; r) with ω 6= G(E; r). In this talk, on the base of the above results, we will investigate the Levi problem in the Riemann domain (ω, ϕ).
References [1] S. Dineen, Complex Analysis in Locally Convex Spaces, North-Holland Math. Studies, 57(1981). [2] S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Springer Monographs Math., Springer Verlag, London(1999). [3] M. Nishihara, On a pseudoconvex domain spread over a complex projective space induced from a complex Banach space with a Schauder basis, J. Math. Soc. Japan, 39(1987), 701-717. [4] M. Nishihara, On the indicator of growth of entire functions of exponential type in infinite dimensional spaces and the Levi problem in infinite dimensional projective spaces, Portugaliae Mathematica, 52(1995), 61-94. [5] M. Nishihara, Riemann domains over Banach-Grassmann manifolds, Asian-European Journal of Mathematics, 2(2009), 503-520. [6] K. Oka, Sur les fonctions analytiques de plusieurs variables complex, VI. Domaines pseudoconvexes, Tˆ ohoku Math. J., 49(1942), 15-52. [7] K. Oka, Sur les fonctions analytiques de plusieurs variables complex, IX, Domaines finis sans point critique int´ereur, Japan. J. Math., 23(1953), 97-155. [8] Y .T. Siu, Pseudoconvexity and the problem of Levi, Bull. Am. Math. Soc., 84(1978), 481-512.
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