The Dolbeault complex in infinite dimensions I

Lempert, László

doi:10.1090/S0894-0347-98-00266-5

The Dolbeault complex in infinite dimensions I
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by László Lempert

J. Amer. Math. Soc. 11 (1998), 485-520

DOI: https://doi.org/10.1090/S0894-0347-98-00266-5

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Part II: J. Amer. Math. Soc. (1999), 775-793

Abstract:

In this paper we introduce certain basic notions concerning infinite dimensional complex manifolds, and prove that the Dolbeault cohomology groups of infinite dimensional projective spaces, with values in finite rank vector bundles, vanish. Some applications of such vanishing theorems are discussed; e.g., we classify vector bundles of finite rank over infinite dimensional projective spaces. Finally, we prove a sharp theorem on solving the inhomogeneous Cauchy–Riemann equations on affine spaces.

References

R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, tensor analysis, and applications, 2nd ed., Applied Mathematical Sciences, vol. 75, Springer-Verlag, New York, 1988. MR 960687, DOI 10.1007/978-1-4612-1029-0
Robert Bonic and John Frampton, Smooth functions on Banach manifolds, J. Math. Mech. 15 (1966), 877–898. MR 0198492
M. J. Bowick and S. G. Rajeev, The holomorphic geometry of closed bosonic string theory and $\textrm {Diff}\,S^1/S^1$, Nuclear Phys. B 293 (1987), no. 2, 348–384. MR 908048, DOI 10.1016/0550-3213(87)90076-9
Jean-Luc Brylinski, Loop spaces, characteristic classes and geometric quantization, Progress in Mathematics, vol. 107, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1197353, DOI 10.1007/978-0-8176-4731-5
W. Barth and A. Van de Ven, A decomposability criterion for algebraic $2$-bundles on projective spaces, Invent. Math. 25 (1974), 91–106. MR 379515, DOI 10.1007/BF01389999
G. Coeuré, Les équations de Cauchy-Riemann sur un espace de Hilbert, manuscript.
J.-F. Colombeau and B. Perrot, L’équation $\bar \partial$ dans les ouverts pseudo-convexes des espaces DFN, Bull. Soc. Math. France 110 (1982), no. 1, 15–26 (French, with English summary). MR 662127, DOI 10.24033/bsmf.1951
V. I. Danilov, Cohomology of algebraic varieties in Encyclopaedia of Mathematical Sciences, Algebraic Geometry II (I. R. Shafarevich, ed.), Springer, Berlin, 1996.
Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
Seán Dineen, Cousin’s first problem on certain locally convex topological vector spaces, An. Acad. Brasil. Ci. 48 (1976), no. 1, 11–12. MR 438376
Seán Dineen, Complex analysis in locally convex spaces, Notas de Matemática [Mathematical Notes], vol. 83, North-Holland Publishing Co., Amsterdam-New York, 1981. MR 640093
Leon Ehrenpreis, A new proof and an extension of Hartogs’ theorem, Bull. Amer. Math. Soc. 67 (1961), 507–509. MR 131663, DOI 10.1090/S0002-9904-1961-10661-7
Hans Grauert, Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann. 135 (1958), 263–273 (German). MR 98199, DOI 10.1007/BF01351803
Richard S. Hamilton, Deformation of complex structures on manifolds with boundary. I. The stable case, J. Differential Geometry 12 (1977), no. 1, 1–45. MR 477158
Richard S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65–222. MR 656198, DOI 10.1090/S0273-0979-1982-15004-2
Christopher J. Henrich, The $\bar \partial$ equation with polynomial growth on a Hilbert space, Duke Math. J. 40 (1973), 279–306. MR 320745
C. Denson Hill, What is the notion of a complex manifold with a smooth boundary?, Algebraic analysis, Vol. I, Academic Press, Boston, MA, 1988, pp. 185–201. MR 992454
T. Honda, The existence of solutions of the $\overline {\partial }$-problem on an infinite dimensional Riemann domain, in Geometric Analysis (J. Noguchi et al. eds ), World Sci. Publ. Co., Singapore, 1996, pp. 249–260.
Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR 1045639
Hans Heinrich Keller, Differential calculus in locally convex spaces, Lecture Notes in Mathematics, Vol. 417, Springer-Verlag, Berlin-New York, 1974. MR 0440592, DOI 10.1007/BFb0070564
Garo K. Kiremidjian, A direct extension method for CR structures, Math. Ann. 242 (1979), no. 1, 1–19. MR 537322, DOI 10.1007/BF01420478
Kunihiko Kodaira, Complex manifolds and deformation of complex structures, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 283, Springer-Verlag, New York, 1986. Translated from the Japanese by Kazuo Akao; With an appendix by Daisuke Fujiwara. MR 815922, DOI 10.1007/978-1-4613-8590-5
J. La Fontaine, Épilogue, Fables, Livre VI, Le livre de poche, Paris, 1972.
Claude LeBrun, A Kähler structure on the space of string worldsheets, Classical Quantum Gravity 10 (1993), no. 9, L141–L148. MR 1237011
László Lempert, Embeddings of three-dimensional Cauchy-Riemann manifolds, Math. Ann. 300 (1994), no. 1, 1–15. MR 1289827, DOI 10.1007/BF01450472
László Lempert, Loop spaces as complex manifolds, J. Differential Geom. 38 (1993), no. 3, 519–543. MR 1243785
László Lempert, The Virasoro group as a complex manifold, Math. Res. Lett. 2 (1995), no. 4, 479–495. MR 1355709, DOI 10.4310/MRL.1995.v2.n4.a8
Ewa Ligocka, Levi forms, differential forms of type $(0,1)$ and pseudoconvexity in Banach spaces, Ann. Polon. Math. 33 (1976), no. 1-2, 63–69. MR 427685, DOI 10.4064/ap-33-1-63-69
Pierre Mazet, Analytic sets in locally convex spaces, North-Holland Mathematics Studies, vol. 89, North-Holland Publishing Co., Amsterdam, 1984. Notas de Matemática [Mathematical Notes], 93. MR 756238
Peter W. Michor, Manifolds of differentiable mappings, Shiva Mathematics Series, vol. 3, Shiva Publishing Ltd., Nantwich, 1980. MR 583436
J. Milnor, Remarks on infinite-dimensional Lie groups, Relativity, groups and topology, II (Les Houches, 1983) North-Holland, Amsterdam, 1984, pp. 1007–1057. MR 830252
M. Mitrea, F. Sabac, Pompeiu’s integral representation formula. History and mathematics, manuscript.
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
Reinhold Meise and Dietmar Vogt, Counterexamples in holomorphic functions on nuclear Fréchet spaces, Math. Z. 182 (1983), no. 2, 167–177. MR 689294, DOI 10.1007/BF01175619
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
P. Raboin, Le problème du $\overline {\partial }$ sur un espace de Hilbert, Bull. Soc. Math. Fr. 107 (1979), 225–240.
Anuradha Jaiswal and Bijendra Singh, On approximation problem in non-Archimedean quasi-normed spaces, Math. Sem. Notes Kobe Univ. 3 (1975), no. 1, paper no. XV, 2. MR 428015
S. Mac Lane, Locally small categories and the foundations of set theory, Infinitistic Methods (Proc. Sympos. Foundations of Math., Warsaw, 1959), Pergamon, Oxford; Państwowe Wydawnictwo Naukowe, Warsaw, 1961, pp. 25–43. MR 0191941
Henri Skoda, $d^{\prime \prime }$-cohomologie à croissance lente dans C$^{n}$, Ann. Sci. École Norm. Sup. (4) 4 (1971), 97–120 (French). MR 287034, DOI 10.24033/asens.1207
A.N. Tjurin, Vector bundles of finite rank over infinite varieties, Math. USSR Izvestija 10 (1976), 1187–1204.
H. Toruńczyk, Smooth partitions of unity on some non-separable Banach spaces, Studia Math. 46 (1973), 43–51. MR 339255, DOI 10.4064/sm-46-1-43-51
A. Van de Ven, On uniform vector bundles, Math. Ann. 195 (1972), 245–248.