Abstract
We use the Leray spectral sequence for the sheaf cohomology groups with compact supports to obtain a vanishing result. The stalks of sheaves \(R^{\bullet }\phi _{!}\mathcal {O}\) for the structure sheaf \(\mathcal {O}\) on the total space of a holomorphic fiber bundle \(\phi \) has canonical topology structures. Using the standard Čech argument we prove a density lemma for QDFS-topology on this stalks. In particular, we obtain a vanishing result for holomorphic fiber bundles with Stein fibers. Using Künnet formulas, properties of an inductive topology (with respect to the pair of spaces) on the stalks of the sheaf \(R^{1}\phi _{!}\mathcal {O}\) and a cohomological criterion for the Hartogs phenomenon we obtain the main result on the Hartogs phenomenon for the total space of holomorphic fiber bundles with (1, 0)-compactifiable fibers.
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References
Andersson, M., Samuelsson, H.: Koppelman Formulas and the \(\bar{\partial }\)-Equation on an Analytic Space. Institut Mittag–Leffler preprint Series, Vol. 31 (2008)
Andreotti, A., Hill, D.: E.E. Levi convexity and the Hans Levy problem. Part I: reduction to vanishing theorems. Ann. Sc. Norm. Super Pisa 26, 325–363 (1972)
Bănică, C., Stănăşilă, O.: Algebraic Methods in the Global Theory of Complex Spaces. Wiley, New York (1976)
Bredon, G.E.: Sheaf Theory. Springer, New York (1997)
Coltoiu, M., Ruppenthal, J.: On Hartogs extension theorem on \((n-1)\)-complete complex spaces. J. Reine Angew. Math. (Crelles J.) 637, 41–47 (2009). https://doi.org/10.1515/CRELLE.2009.089
Demailly, J.-P.: Complex Analytic and Differential Geometry (2012). https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf
Dwilewicz, R.: Holomorphic extensions in complex fiber bundles. J. Math. Anal. Appl. 322, 556–565 (2006). https://doi.org/10.1016/j.jmaa.2005.09.044
Freudenthal, H.: Über die Enden topologischer Räume und Gruppen. Math. Z. 33, 692–713 (1931). https://doi.org/10.1007/BF01174375
Forstnerič, F.: Stein Manifolds and Holomorphic Mappings. Springer, Berlin, Heidelberg (2011)
Feklistov, S.: The Hartogs extension phenomenon in almost homogeneous algebraic varieties. Mat. Sb. 213(12), 109–136 (2022). https://doi.org/10.4213/sm9677
Feklistov, S., Shchuplev, A.: The Hartogs extension phenomenon in toric varieties. J. Geom. Anal. 31, 12034–12052 (2021). https://doi.org/10.1007/s12220-021-00710-4
Gunning, R.C., Rossi, H.: Analytic Functions of Several Complex Variables. Prentice-Hall Inc, Englewood Cliffs (1965)
Grothendieck, A., Chaljub, O.: Topological Vector Spaces. Gordon and Breach, Notes on mathematics and its applications (1973)
Harvey, R.: The theory of hyperfunctions on totally real subsets of complex manifolds with applications to extension problems. Am. J. Math. 91, 853–873 (1969). https://doi.org/10.2307/2373307
Iversen, B.: Cohomology of Sheaves. Universitext, Springer, Berlin, Heidelberg (1986)
Kaup, L.: Eine Künnethformel für Fréchetgarben. Math. Z. 97(2), 158–168 (1967)
Marciniak, M.A.: Holomorphic extensions in toric varieties. Doctoral Dissertations, Ph. D. in Mathematics, Missouri (2009)
Marciniak, M.A.: Holomorphic extensions in smooth toric surfaces. J. Geom. Anal. 22, 911–933 (2012). https://doi.org/10.1007/s12220-011-9219-7
Merker, J., Porten, E.: The Hartogs extension theorem on \((n-1)\)-complete complex spaces. J. Reine Angew. Math. 637, 23–39 (2009). https://doi.org/10.1515/CRELLE.2009.088
Øvrelid, N., Vassiliadou, S.: Hartogs extension theorems on Stein spaces. J. Geom. Anal. 20, 817–836 (2010). https://doi.org/10.1007/s12220-010-9134-3
Oda, T.: Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric Varieties, Springer, Berlin (1988)
Peschke, G.: The theory of ends. Nieuw Arch. Wiskunde 8, 1–12 (1990)
Pietsch, A., Ruckle, W.H.: Nuclear Locally Convex Spaces. Springer, Berlin, Heidelberg (1972)
Rao, S., Yang, S., Yang, X.: Dolbeault cohomologies of blowing up complex manifolds. J. Math. Pures Appl. 130, 68–92 (2019). https://doi.org/10.48550/arXiv.1712.06749
Serre, J. P.: Quelques problemes globaux relatifs aux varietes de Stein. In: Coll. Plus. Var. Bruxelles, pp. 57–68 (1953)
Schaefer, H.H., Wolff, M.P.: Topological Vector Spaces. Graduate Texts in Mathematics, Springer, New York (1999)
Vîjîitu, V.: On Hartogs’ extension. Ann. Mat. Pura Appl. 201, 487–498 (2022). https://doi.org/10.1007/s10231-021-01125-2
Voisin, C.: Hodge Theory and Complex Algebraic Geometry II. Cambridge Studies in Advanced Mathematics, vol. 77. Cambridge University Press, Cambridge (2009)
Acknowledgements
This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation (Agreement No. 75-02-2023-936).
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Feklistov, S. Holomorphic extension in holomorphic fiber bundles with (1, 0)-compactifiable fiber. Annali di Matematica (2023). https://doi.org/10.1007/s10231-023-01412-0
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DOI: https://doi.org/10.1007/s10231-023-01412-0
Keywords
- Hartogs phenomenon
- Holomorphic extension
- Holomorphic fiber bundle
- (1, 0)-compactifiable complex manifold