Abstract
Flag domains are open orbits of noncompact real forms of complex semisimple Lie groups acting on flag manifolds. To each flag domain one can associate a compact complex manifold called the base cycle. The ampleness of the normal bundle of the base cycle in a flag domain measures the concavity near the base cycle. In this paper we compute the ampleness of normal bundles of base cycles in flag domains in various cases, including flag domains in the full flag manifolds G/B when G is classical, and period domains parameterizing polarized Hodge structures with fixed Hodge numbers.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Baston, R.J., Eastwood, M.G.: The Penrose transform. Its interaction with representation theory. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1989). xvi+213 pp
Carlson, J., Müller-Stach, S., Peters, C.: Period mappings and period domains. Second edition of [MR2012297]. Cambridge Studies in Advanced Mathematics, 168. Cambridge University Press, Cambridge (2017). xiv+562 pp. ISBN: 978-1-316-63956-6; 978-1-108-42262-8
Fels, G., Huckleberry, A., Wolf, J.: Cycle spaces of flag domains. A complex geometric viewpoint. Progress in Mathematics 245. Birkhäuser Boston, Inc., Boston, MA (2006)
Hayama, T., Huckleberry, A., Latif, Q.: Pseudoconcavity of flag domains: the method of supporting cycles. Math. Ann. 375, 671–685 (2019)
Hong, J., Huckleberry, A., Seo, A.: Normal bundles of cycles in flag domains. São Paulo J. Math. Sci. 12(2), 278–289 (2018)
Huckleberry, A.: Hyperbolicity of cycle spaces and automorphism groups of flag domains. Amer. J. Math. 135, 291–310 (2013)
Huckleberry, A.T., Simon, A., Barlet, D.: On cycle spaces of flag domains of \(SL_n(\mathbb{R} )\). J. Reine Angew. Math. 541, 171–208 (2001)
Huckleberry, A.T., Wolf, J.A.: Cycle spaces of real forms of \(SL_n(\mathbb{C})\). Complex geometry (Göttingen, 2000), pp. 111–133. Springer, Berlin (2002)
Knapp, A.W.: Lie groups beyond an introduction. Second edition. Progress in Mathematics, 140. Birkhäuser Boston, Inc., Boston, MA (2002). xviii+812 pp. ISBN: 0-8176-4259-5 22-01
Matsumura, S.-i.: Asymptotic cohomology vanishing and a converse to the Andreotti-Grauert theorem on surfaces. Ann. Inst. Fourier (Grenoble) 63(6), 2199–2221 (2013)
Onishchik, A.L., Vinberg, É.B.: Lie groups and algebraic groups. Translated from the Russian and with a preface by D. A. Leites. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin (1990). xx+328 pp
Snow, D.: On the ampleness of homogeneous vector bundles. Trans. Amer. Math. Soc. 294(2), 585–594 (1986)
Snow, D.: Homogeneous vector bundles, available at https://www3.nd.edu/~snow/. Access year (2023)
Sommese, A.J.: Submanifolds of Abelian varieties. Math. Ann. 233(3), 229–256 (1978)
Sommese, A.J.: A convexity theorem. Singularities, Part 2 (Arcata, Calif., 1981), 497–505, Proc. Sympos. Pure Math. 40 Amer. Math. Soc., Providence, RI (1983)
Voisin, C.: Hodge theory and complex algebraic geometry I. Translated from the French original by Leila Schneps. Cambridge Studies in Advanced Mathematics, 76. Cambridge University Press, Cambridge (2002). x+322 pp. ISBN: 0-521-80260-1
Acknowledgements
The authors would like to express their gratitude to the anonymous referee for carefully reading the manuscript and providing excellent suggestions for improvement.
Funding
The first author was supported by the Institute for Basic Science (IBS-R032-D1). The second author was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2022R1F1A1063038).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing Interests
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Hong, J., Seo, A. Ampleness of Normal Bundles of Base Cycles in Flag Domains. Transformation Groups (2023). https://doi.org/10.1007/s00031-023-09831-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00031-023-09831-2