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Transactions of the American Mathematical Society

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Semiglobal results for $ \overline\partial$ on complex spaces with arbitrary singularities, Part II

Authors: Nils Øvrelid and Sophia Vassiliadou
Journal: Trans. Amer. Math. Soc. 363 (2011), 6177-6196
MSC (2010): Primary 32B10, 32J25, 32W05
Published electronically: July 14, 2011
MathSciNet review: 2833549
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Abstract: We obtain some $ L^2$-results for $ \overline\partial$ on forms that vanish to high order on the singular set of a complex space. As a consequence of our main theorem we obtain weighted $ L^2$-solvability results for compactly supported $ \overline\partial$-closed $ (p,q)$-forms $ (0\le p\le n, 1\le q< n)$ on relatively compact subdomains $ \Omega$ of the complex space that satisfy $ H^{n-q}(\Omega, \mathcal{S})=0=H^{n-q+1}(\Omega, \mathcal{S})$ for every coherent $ \mathcal{O}_X$-module $ \mathcal{S}$. The latter result can be used to give an alternate proof of a theorem of Merker and Porten.

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  • 1. J.M. Aroca, H. Hironaka and J.L. Vicente, Desingularization theorems, Mem. Math. Inst. Jorge Juan, No. 30, Madrid, 1977. MR 480502 (80h:32027)
  • 2. E. Bierstone and P. Milman, Canonical desingularization in characteristic zero by blowing-up the maximum strata of a local invariant, Inventiones Math., 128, no. 2 (1997), 207-302. MR 1440306 (98e:14010)
  • 3. M. Coltoiu, A supplement to a theorem of Merker and Porten: A short proof of Hartogs extension theorem for $ (n-1)$-complete complex spaces, preprint available at, arxiv:0811.2352.
  • 4. M. Coltoiu and J. Ruppenthal, On Hartogs' extension theorem on $ (n-1)$-complete complex spaces, J. Reine Angew. Math, vol. 637 (2009), 41-47. MR 2599080
  • 5. J.E. Fornæss, N. Øvrelid and S. Vassiliadou, Local $ L^2$-results for $ \overline\partial$: The isolated singularities case, Internat. J. of Math., 16, no. 4 (2005), 387-418. MR 2133263 (2006b:32048)
  • 6. J. E. Fornæss, N. Øvrelid and S. Vassiliadou, Semiglobal results for $ \overline\partial$-on a complex space with arbitrary singularities, Proc. of Amer. Math. Soc., vol. 133, no. 8 (2005), 2377-2386. MR 2138880 (2006c:32048)
  • 7. H. Grauert, Ein Theorem der analytischen Garbentheorie und die Modulraüme komplexer Structuren, Publ.Math. Inst. Hautes Etud. Sc., no. 5 (1960), 5-64. MR 0121814 (22:12544)
  • 8. H. Grauert, Über Modificationen und exzeptionelle analytische Mengen, Math. Ann., 146, (1962), 331-368. MR 0137127 (25:583)
  • 9. H. Grauert, On Levi's problem and the imbedding of real-analytic manifolds, The Annals of Math., vol. 68, no. 2 (1958), 460-472. MR 0098847 (20:5299)
  • 10. H. Grauert, T. Peternell and R. Remmert, Several Complex Variables VII, Encyclopaedia of Mathematical Sciences, Volume 74, 1994, Springer-Verlag. MR 1326617 (96k:32001)
  • 11. H. Grauert and R. Remmert, Coherent Analytic Sheaves, Grundlehren der mathematischen Wissenschaften 265, Springer-Verlag, 1984. MR 755331 (86a:32001)
  • 12. R. Gunning Introduction to Holomorphic Functions of Several Variables, I, II, III, Wadsworth and Brooks/Cole Mathematics Series, 1990. MR 1052649 (92b:32001a), MR 1057177 (92b:32001b), MR 1059457 (92b:32001c)
  • 13. L. Hörmander, An Introduction to Complex Analysis in Several Variables, North-Holland Mathematical Library, 3rd edition, 1990. MR 1045639 (91a:32001)
  • 14. L. Hörmander, The Analysis of Linear Partial Differential operators I, 2nd. edition, Springer-Verlag, 1990. MR 1065993 (91m:35001a)
  • 15. N. Jacobson, Basic Algebra II, 2nd edition, W. H. Freeman and Company, New York, 1989. MR 1009787 (90m:00007)
  • 16. J. Merker and E. Porten, The Hartogs extension theorem on $ (n-1)$-complete complex spaces, J. Reine Angew. Math, vol. 637 (2009), 23-39. MR 2599079
  • 17. N. Øvrelid and S. Vassiliadou, Hartogs extension theorems on Stein spaces, J. Geom. Anal. 20, (2010), no. 4, 817-836. MR 2683769
  • 18. J. Ruppenthal, A $ \overline{\partial}$ theoretical proof of Hartogs extension theorem on $ (n-1)$-complete spaces, preprint available at, arxiv:0811.1963.
  • 19. Y. T. Siu, Analytic sheaf cohomology groups of dimension $ n$ of $ n$-dimensional complex spaces, Trans. Amer. Math. Soc., 143 (1969), 77-94. MR 0252684 (40:5902)
  • 20. K. Takegoshi, Relative vanishing theorems in analytic spaces, Duke Math. Journ., 52, no. 1 (1985), 273-279. MR 791302 (86i:32049)

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Additional Information

Nils Øvrelid
Affiliation: Department of Mathematics, University of Oslo, P.B. 1053 Blindern, Oslo, N-0316 Norway

Sophia Vassiliadou
Affiliation: Department of Mathematics, Georgetown University, Washington, DC 20057

Keywords: Cauchy-Riemann equation, singularity, cohomology groups
Received by editor(s): June 19, 2009
Published electronically: July 14, 2011
Additional Notes: The research of the second author was partially supported by NSF grant DMS-0712795
Article copyright: © Copyright 2011 American Mathematical Society

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