Details

Stefano Marini ^[a]

On finitely Levi non degenerate homogeneous \(CR\) manifolds

Pages: 353-372
Received: 26 November 2021
Accepted in revised form: 22 April 2022
Mathematics Subject Classification: Primary: 32V35, 32V40, Secondary: 17B22, 17B10.
Keywords: Lie pair, \(CR\) algebra, Lie algebra extension, Levi degeneracy.
Author address:
[a]: Università di Parma, Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Parma, Italy

Abstract: A \(CR\) manifold M is a differentiable manifold together with a complex subbundle of the complexification of its tangent bundle, which is formally integrable and has zero intersection with its conjugate bundle. A fundamental invariant of a \(CR\) manifold \(\textsf{M}\) is its vector-valued Levi form. A Levi non degenerate \(CR\) manifold of order \(k \geq 1\) has non degenerate Levi form in a higher order sense. For a (locally) homogeneous manifold Levi non degeneracy of order \(k\) can be described in terms of its \(CR\) algebra, i.e. a pair of Lie algebras encoding the structure of (locally) homogeneous \(CR\) manifolds. I will give an introduction to these topics presenting some recent results.

References

[1]: A. Altomani, C. Medori and M. Nacinovich, The CR structure of minimal orbits in complex flag manifolds, J. Lie Theory 16 (2006), no. 3, 483-530. MR2248142
[2]: A. Altomani, C. Medori and M. Nacinovich, On the topology of minimal orbits in complex flag manifolds, Tohoku Math. J. (2) 60 (2008), no. 3, 403-422. MR2453731
[3]: A. Altomani, C. Medori and M. Nacinovich, Orbits of real forms in complex flag manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), no. 1, 69-109. MR2668874
[4]: N. Bourbaki, Lie groups and Lie algebras. Chapters 4-6, Translated from the 1968 French original by Andrew Pressley, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002, MR1890629
[5]: N. Bourbaki, Lie groups and Lie algebras. Chapters 7-9, Translated from the 1975 and 1982 French originals by Andrew Pressley, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2005. MR2109105
[6]: R. Bremigan and J. Lorch, Orbit duality for flag manifolds, Manuscripta Math. 109 (2002), no. 2, 233-261. MR1935032
[7]: G. Fels, Locally homogeneous finitely nondegenerate CR-manifolds, Math. Res. Lett. 14 (2007), no. 6, 893-922. MR2357464
[8]: G. Fels, A. T. Huckleberry and J. A. Wolf, Cycle spaces of flag domains, A complex geometric viewpoint, Progr. Math., 245, Birkhäuser Boston Inc., Boston, MA, 2006. MR2188135
[9]: M. Freeman, Local biholomorphic straightening of real submanifolds, Ann. of Math. 106 (1977), 319-352. MR0463480
[10]: M. Freeman, Real submanifolds with degenerate Levi form, Several complex variables, Part 1, Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1975, vol. XXX, Amer. Math. Soc., 1977, 141-147. MR0457767
[11]: A. N. García and C. U. Sánchez, On extrinsic symmetric CR-structures on the manifolds of complete flags, Beiträge Algebra Geom. 45 (2004), no. 2, 401-414. MR2093174
[12]: S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Grad. Stud. Math., 34, American Mathematical Society, Providence, RI, 2001. MR1834454
[13]: A. Lotta and M. Nacinovich, On a class of symmetric CR manifolds, Adv. Math. 191 (2005), no. 1, 114-146. MR2102845
[14]: S. Marini, C. Medori, M. Nacinovich and A. Spiro, On transitive contact and \(CR\) algebras, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 20 (2020), 771-795. MR4105918
[15]: S. Marini, C. Medori and M. Nacinovich, \(\mathcal{L}\)-prolongations of graded Lie algebras, Geom. Dedicata 208 (2020), 61-88. MR4142917
[16]: S. Marini, C. Medori and M. Nacinovich, On some classes of \(\mathbb{Z}\)-graded Lie algebras, Abh. Math. Semin. Univ. Hambg. 90 2020, 45-71. MR4131919
[17]: S. Marini, C. Medori and M. Nacinovich, Higher order Levi forms on homogeneous CR manifolds, Math. Z. 299 (2021), no. 1-2, 563-589. MR4311613
[18]: S. Marini and M. Nacinovich, Orbits of real forms, Matsuki duality and \(CR\)-cohomology, Complex and symplectic geometry, Springer INdAM Ser., vol. 21, Springer, Cham, 2017, 149-162. MR3645313
[19]: S. Marini and M. Nacinovich, Mostow's fibration for canonical embeddings of compact homogeneous \(CR\) manifolds, Rend. Semin. Mat. Univ. Padova 140 (2018), 1-43. MR3881753
[20]: C. Medori and M. Nacinovich, Complete nondegenerate locally standard CR manifolds, Math. Ann. 317 (2000), no. 3, 509-526. MR1776115
[21]: C. Medori and M. Nacinovich, Algebras of infinitesimal CR automorphisms, J. Algebra 287 (2005), no. 1, 234-274. MR2134266
[22]: A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. 65 (1957), 391-404. MR0088770
[23]: P. J. Hilton and U. Stammbach, A course in homological algebra, Graduate Texts in Mathematics, vol. 4, Springer-Verlag, New York-Berlin, 1971. MR0346025
[24]: M. Tinkham, Group theory and quantum mechanics, McGraw-Hill, New York-Toronto, 1964. MR0198828
[25]: J. A. Wolf, The action of a real semisimple group on a complex flag manifold, I, Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc. 75 (1969), 1121-1237. MR0251246