Abstract
A flag domain of a real from \(G_0\) of a complex semismiple Lie group G is an open \(G_0\)-orbit D in a (compact) G-flag manifold. In the usual way one reduces to the case where \(G_0\) is simple. It is known that if D possesses non-constant holomorphic functions, then it is the product of a compact flag manifold and a Hermitian symmetric bounded domain. This pseudoconvex case is rare in the geography of flag domains. Here it is shown that otherwise, i.e., when \(\mathscr {O}(D)\cong \mathbb {C}\), the flag domain D is pseudoconcave. In a rather general setting the degree of the pseudoconcavity is estimated in terms of root invariants. This estimate is explicitly computed for domains in certain Grassmannians.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andreotti, A.: Théorèmes de dépendance algébrique sur les espaces complexes pseudo-concaves. Bull. Soc. Math. France 91, 1–38 (1963)
Andreotti, A., Grauert, H.: Théorèmes de finitude pour la cohomologie des espaces complexes. Bull. Soc. Math. France 90, 193–259 (1962)
Burns, D., Haverscheid, S., Hind, R.: The geometry of Grauert tubes and complexification of symmetric spaces. Duke. J. Math. 118, 465–491 (2003)
G, Fels, Huckleberry, A., Wolf, A.: Flag Domains: A Complex Geometric Viewpoint, Progress in Mathematics. Springer, Boston (2005)
Grauert, H., Remmert, R.: Coherent Analytic Sheaves Grundlehren der math. Wiss. Springer, New York (1984)
Grauert, H., Remmert, R.: Theory of Stein Spaces, Classics in Mathematics. Springer, New York (1979)
Hayama, T.: Cycle connectivity and pseudoconcavity of flag domains (arXiv:1501.01768)
Hayama, T., Huckleberry, A., Latif, Q.: Cycle connectivity of flag domains (preprint in preparation)
Hong, J, Huckleberry, A. and Seo, A.: Normal bundles of cycles in flag domains (arXiv:1807.07311v1)
Huckleberry, A.: Remarks on homogeneous manifolds satisfying Levi-conditions, Bollettino U.M.I. (9) III (2010) 1-23 (arXiv:1003:5971)
Huckleberry, A.: Cycle connectivity and automorphism groups of flag domains, In: Mason, G., Penkov, I., Wolf, JA. (eds.) Developments and Retrospectives in Lie Theory, Geometric and Analytic Methods, Series: Developments in Mathematics, Vol. 37, VIII, Springer, Berlin (2014)
Kollar, J.: Neighborhoods of subvarieties in homogeneous spaces. Contemp. Math. 647, 91–107 (2015)
Latif, Q.: On the pseudoconcavity of flag domains, Jacobs University Thesis, June 2017
Wolf, J.A.: The action of a real semisimple Lie group on a complex manifold, I: orbit structure and holomorphic arc components. Bull. Am. Math. Soc. 75, 1121–1237 (1969)
Acknowledgements
The work was supported by Deutsche Forschungsgemeinschaft (Grant No. Schwerpunkt 1388).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Ngaiming Mok.
The first author was supported by JSPS KAKENHI Grant Number 16K17576. The third author was supported in the project by the a grant in the DFG-Schwerpunkt 1388, “Darstellungstheorie”. Part of his Jacobs University thesis ([13]), which was guided by the second author and strongly influenced by the first author, was devoted to the results in the present paper. The first and second author’s joint work was supported by funds from the Korean Institute of Advanced Study (KIAS) and the NSF during their visits to Seoul, respectively Berkeley. We are all thankful for this support.
Rights and permissions
About this article
Cite this article
Hayama, T., Huckleberry, A. & Latif, Q. Pseudoconcavity of flag domains: the method of supporting cycles. Math. Ann. 375, 671–685 (2019). https://doi.org/10.1007/s00208-018-1737-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-018-1737-1