Invariant holomorphic foliations on Kobayashi hyperbolic homogeneous manifolds
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- by Filippo Bracci, Andrea Iannuzzi and Benjamin McKay PDF
- Proc. Amer. Math. Soc. 144 (2016), 1619-1629 Request permission
Abstract:
Let $M$ be a Kobayashi hyperbolic homogeneous manifold. Let $\mathcal F$ be a holomorphic foliation on $M$ invariant under a transitive group $G$ of biholomorphisms. We prove that the leaves of $\mathcal F$ are the fibers of a holomorphic $G$-equivariant submersion $\pi \colon M \to N$ onto a $G$-homogeneous complex manifold $N$. We also show that if $\mathcal Q$ is an automorphism family of a hyperbolic convex (possibly unbounded) domain $D$ in $\mathbb {C}^n$, then the fixed point set of $\mathcal Q$ is either empty or a connected complex submanifold of $D$.References
- Marco Abate, Iteration theory of holomorphic maps on taut manifolds, Research and Lecture Notes in Mathematics. Complex Analysis and Geometry, Mediterranean Press, Rende, 1989. MR 1098711
- Paul Baum and Raoul Bott, Singularities of holomorphic foliations, J. Differential Geometry 7 (1972), 279–342. MR 377923
- Alexandre Behague and Bruno Scárdua, Foliations invariant under Lie group transverse actions, Monatsh. Math. 153 (2008), no. 4, 295–308. MR 2394552, DOI 10.1007/s00605-008-0523-7
- S. Bochner, Compact groups of differentiable transformations, Ann. of Math. (2) 46 (1945), 372–381. MR 13161, DOI 10.2307/1969157
- Filippo Bracci and Alberto Saracco, Hyperbolicity in unbounded convex domains, Forum Math. 21 (2009), no. 5, 815–825. MR 2560392, DOI 10.1515/FORUM.2009.039
- Marco Brunella, Birational geometry of foliations, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2004. MR 2114696
- Franc Forstneric, Interpolation by holomorphic automorphisms and embeddings in $\textbf {C}^n$, J. Geom. Anal. 9 (1999), no. 1, 93–117. MR 1760722, DOI 10.1007/BF02923090
- Laura Geatti, Holomorphic automorphisms of the tube domain over the Vinberg cone, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 80 (1986), no. 5, 283–291 (1987) (English, with Italian summary). MR 977139
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34, American Mathematical Society, Providence, RI, 2001. Corrected reprint of the 1978 original. MR 1834454, DOI 10.1090/gsm/034
- Hideyuki Ishi, A torus subgroup of the isotropy group of a bounded homogeneous domain, Manuscripta Math. 130 (2009), no. 3, 353–358. MR 2545522, DOI 10.1007/s00229-009-0291-2
- Soji Kaneyuki, Homogeneous bounded domains and Siegel domains, Lecture Notes in Mathematics, Vol. 241, Springer-Verlag, Berlin-New York, 1971. MR 0338467
- Shoshichi Kobayashi, Hyperbolic complex spaces, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 318, Springer-Verlag, Berlin, 1998. MR 1635983, DOI 10.1007/978-3-662-03582-5
- Christian Miebach, Quotients of bounded homogeneous domains by cyclic groups, Osaka J. Math. 47 (2010), no. 2, 331–352. MR 2722364
- Subhashis Nag, Hyperbolic manifolds admitting holomorphic fiberings, Bull. Austral. Math. Soc. 26 (1982), no. 2, 181–184. MR 683651, DOI 10.1017/S0004972700005694
- Kazufumi Nakajima, Homogeneous hyperbolic manifolds and homogeneous Siegel domains, J. Math. Kyoto Univ. 25 (1985), no. 2, 269–291. MR 794987, DOI 10.1215/kjm/1250521109
- H. L. Royden, Holomorphic fiber bundles with hyperbolic fiber, Proc. Amer. Math. Soc. 43 (1974), 311–312. MR 338465, DOI 10.1090/S0002-9939-1974-0338465-0
- D. Tischler, On fibering certain foliated manifolds over $S^{1}$, Topology 9 (1970), 153–154. MR 256413, DOI 10.1016/0040-9383(70)90037-6
- V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Graduate Texts in Mathematics, vol. 102, Springer-Verlag, New York, 1984. Reprint of the 1974 edition. MR 746308, DOI 10.1007/978-1-4612-1126-6
Additional Information
- Filippo Bracci
- Affiliation: Dipartimento Di Matematica, Università di Roma “Tor Vergata”, Via Della Ricerca Scientifica 1, 00133 Roma, Italy
- MR Author ID: 631111
- Email: fbracci@mat.uniroma2.it
- Andrea Iannuzzi
- Affiliation: Dipartimento Di Matematica, Università di Roma “Tor Vergata”, Via Della Ricerca Scientifica 1, 00133 Roma, Italy
- Email: iannuzzi@mat.uniroma2.it
- Benjamin McKay
- Affiliation: University College Cork, National University of Ireland, Cork, Ireland
- Email: b.mckay@ucc.ie
- Received by editor(s): March 3, 2015
- Received by editor(s) in revised form: April 23, 2015
- Published electronically: July 30, 2015
- Additional Notes: The first author was supported by the ERC grant “HEVO - Holomorphic Evolution Equations” n. 277691.
- Communicated by: Franc Forstneric
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 1619-1629
- MSC (2010): Primary 37F75; Secondary 32Q45, 32M10
- DOI: https://doi.org/10.1090/proc/12817
- MathSciNet review: 3451238