Previous |  Up |  Next

Article

Title: Exterior differential systems, Lie algebra cohomology, and the rigidity of homogenous varieties (English)
Author: Landsberg, Joseph M.
Language: English
Journal: Archivum Mathematicum
ISSN: 0044-8753 (print)
ISSN: 1212-5059 (online)
Volume: 44
Issue: 5
Year: 2008
Pages: 419-447
Summary lang: English
.
Category: math
.
Summary: These are expository notes from the 2008 Srní Winter School. They have two purposes: (1) to give a quick introduction to exterior differential systems (EDS), which is a collection of techniques for determining local existence to systems of partial differential equations, and (2) to give an exposition of recent work (joint with C. Robles) on the study of the Fubini-Griffiths-Harris rigidity of rational homogeneous varieties, which also involves an advance in the EDS technology. (English)
Keyword: projective rigidity
Keyword: exterior differential systems
Keyword: Lie algebra cohomology
Keyword: homogeneous varieties
MSC: 14M17
MSC: 53A20
MSC: 53C30
idZBL: Zbl 1212.53013
idMR: MR2501577
.
Date available: 2009-01-29T09:16:10Z
Last updated: 2013-09-19
Stable URL: http://hdl.handle.net/10338.dmlcz/127127
.
Reference: [1] Berger, E., Bryant, R., Griffiths, P.: Some isometric embedding and rigidity results for Riemannian manifolds.Proc. Nat. Acad. Sci. U.S.A. 78 (8), part 1 (1981), 4657–4660. Zbl 0468.53040, MR 0627257, 10.1073/pnas.78.8.4657
Reference: [2] Bourbaki, N.: Groupes et algèbres de Lie.Hermann, Paris, 1968. Zbl 0186.33001, MR 0240238
Reference: [3] Brion, M.: Spherical varieties.Proceedings of the International Congress of Mathematicians, Zürich 1994, vol. 1, 2, Birkhäuser, Basel, 1995, pp. 753–760. Zbl 0862.14031, MR 1403975
Reference: [4] Bryant, R.: Metrics with exceptional holonomy.Ann. of Math. (2) 126 (3) (1987), 525–576. Zbl 0637.53042, MR 0916718, 10.2307/1971360
Reference: [5] Bryant, R. L.: Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spaces.Princeton University Press, AM-153, 2005.
Reference: [6] Bryant, R. L., Chern, S. S., Gardner, R. B., Goldschmidt, H. L., Griffiths, P. A.: Exterior differential systems.Mathematical Sciences Research Institute Publications, 18. Springer-Verlag, New York, 1991. Zbl 0726.58002, MR 1083148, 10.1007/978-1-4613-9714-4_5
Reference: [7] Čap, A.: Lie algebra cohomology and overdetermined systems.preprint.
Reference: [8] Čap, A., Schichl, H.: Parabolic geometries and canonical Cartan connections.Hokkaido Math. J. 29 (3) (2000), 453–505. MR 1795487
Reference: [9] Cartan, E.: Sur les variétés de courbure constante d’un espace euclidien ou non euclidien.Bull. Soc. Math. France 47 (1919), 125–160; ; see also pp. 321–432 in 125–160 125–160 and 48 (1920), 132–208; see also pp. 321–432 in Oeuvres Complètes Part 3, Gauthier–Villars, 1955.
Reference: [10] Cartan, E.: Sur les variétés a connexion projective.Bull. Soc. Math. France 52 (1924), 205–241. MR 1504846
Reference: [11] Chern, S. S., Osserman, R.: Remarks on the Riemannian metric of a minimal submanifold.Geometry Symposium, Utrecht 1980, Lecture Notes in Math., Springer, Berlin-New York 894 (1981), 49–90. Zbl 0477.53056, MR 0655419, 10.1007/BFb0096224
Reference: [12] Deligne, P.: La série exceptionnelle des groupes de Lie.C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), 321–326. MR 1378507
Reference: [13] Deligne, P., de Man, R.: The exceptional series of Lie groups.C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), 577–582. Zbl 0910.22009, MR 1411045
Reference: [14] Fubini, G.: Studi relativi all’elemento lineare proiettivo di una ipersuperficie.Rend. Accad. Naz. dei Lincei (1918), 99–106.
Reference: [15] Griffiths, P. A., Harris, J.: Algebraic geometry and local differential geometry.Ann. Sci. École Norm. Sup. (4) 12 (1979), 355–432. Zbl 0426.14019, MR 0559347
Reference: [16] Harvey, R., Lawson, H. B.: Calibrated geometries.Acta Math. 148 (1982), 47–157. Zbl 0584.53021, MR 0666108, 10.1007/BF02392726
Reference: [17] Hilgert, J.: Multiplicity free branching laws for unitary representations.Srní lectures, 2008.
Reference: [18] Hong, J.: Rigidity of singular Schubert varieties in ${\mathrm{G}r}(m,n)$.J. Differential Geom. 71 (1) (2005), 1–22. MR 2191767
Reference: [19] Hong, J.: Rigidity of smooth Schubert varieties in Hermitian symmetric spaces.Trans. Amer. Math. Soc. 359 (5) (2007), 2361–2381. Zbl 1126.14010, MR 2276624, 10.1090/S0002-9947-06-04041-4
Reference: [20] Hwang, J. M., Yamaguchi, K.: Characterization of Hermitian symmetric spaces by fundamental forms.Duke Math. J. 120 (3) (2003), 621–634. Zbl 1053.32012, MR 2030098, 10.1215/S0012-7094-03-12035-9
Reference: [21] Ivey, T., Landsberg, J. M.: Cartan for beginners: differential geometry via moving frames and exterior differential systems.Grad. Stud. Math. 61 (2003), xiv + 378. Zbl 1105.53001, MR 2003610
Reference: [22] Kebekus, S., Peternell, T., Sommese, A., Wiśniewski, J.: Projective contact manifolds.Invent. Math. 142 (1) (2000), 1–15. MR 1784795, 10.1007/PL00005791
Reference: [23] Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem.Ann. of Math. (2) 74 (1961), 329–387. Zbl 0134.03501, MR 0142696, 10.2307/1970237
Reference: [24] Landsberg, J., Robles, C.: Fubini-Griffiths-Harris rigidity and Lie algebra cohomology.preprint arXiv:0707.3410.
Reference: [25] Landsberg, J. M.: Differential-geometric characterizations of complete intersections.J. Differential Geom. 44 (1996), 32–73. Zbl 0873.53007, MR 1420349
Reference: [26] Landsberg, J. M.: On the infinitesimal rigidity of homogeneous varieties.Compositio Math. 118 (1999), 189–201. Zbl 0981.53039, MR 1713310, 10.1023/A:1017161326705
Reference: [27] Landsberg, J. M.: Griffiths-Harris rigidity of compact Hermitian symmetric spaces.J. Differential Geom. 74 (3) (2006), 395–405. Zbl 1107.53036, MR 2269783
Reference: [28] Landsberg, J. M.: Differential geometry of submanifolds of projective space.Symmetries and overdetermined systems of partial differential equations. Eastwood, Michael (ed.) et al., Proceedings of the IMA summer program, Minneapolis, MN, USA, July 17–August 4, 2006. New York, NY: Springer. The IMA Volumes in Mathematics and its Applications 144 (2008), 105–125. Zbl 1148.53009, MR 2384708
Reference: [29] Landsberg, J. M.: Geometry and the complexity of matrix multiplication.Bull. Amer. Math. Soc., New Ser. 45 (2) (2008), 247–284. Zbl 1145.68054, MR 2383305, 10.1090/S0273-0979-08-01176-2
Reference: [30] Landsberg, J. M., Manivel, L.: The projective geometry of Freudenthal’s magic square.J. Algebra 239 (2) (2001), 477–512. Zbl 1064.14053, MR 1832903, 10.1006/jabr.2000.8697
Reference: [31] Landsberg, J. M., Manivel, L.: Construction and classification of complex simple Lie algebras via projective geometry.Selecta Math. 8 (2002), 137–159. Zbl 1073.14551, MR 1890196, 10.1007/s00029-002-8103-5
Reference: [32] Landsberg, J. M., Manivel, L.: Triality, exceptional Lie algebras, and Deligne dimension formulas.Adv. Math. 171 (2002), 59–85. Zbl 1035.17016, MR 1933384, 10.1006/aima.2002.2071
Reference: [33] Landsberg, J. M., Manivel, L.: On the projective geometry of rational homogeneous varieties.Comment. Math. Helv. 78 (1) (2003), 65–100. Zbl 1048.14032, MR 1966752
Reference: [34] Landsberg, J. M., Manivel, L.: Representation theory and projective geometry.Algebraic Transformation Groups and Algebraic Varieties, V. L. Popov (ed.), Encyclopaedia Math. Sci., vol. 132, Springer, 2004, pp. 71–122. Zbl 1145.14316, MR 2090671
Reference: [35] Landsberg, J. M., Manivel, L.: Series of Lie groups.Michigan Math. J. 52 (2) (2004), 453–479. Zbl 1165.17302, MR 2069810, 10.1307/mmj/1091112085
Reference: [36] Landsberg, J. M., Manivel, L.: A universal dimension formula for complex simple Lie algebras.Adv. Math. 201 (2) (2006), 379–407. Zbl 1151.17003, MR 2211533, 10.1016/j.aim.2005.02.007
Reference: [37] Landsberg, J. M., Manivel, L.: The sextonions and $E_{7\frac{1}{2}}$.Adv. Math. 201 (1) (2006), 143–179. MR 2204753
Reference: [38] Landsberg, J. M., Manivel, L.: Legendrian varieties.Asian Math. J. 11 (3) (2007), 341–360. Zbl 1136.14024, MR 2372722, 10.4310/AJM.2007.v11.n3.a1
Reference: [39] Landsberg, J. M., Weyman, J.: On tangential varieties of rational homogeneous varieties.J. London Math. Soc. (2) 76 (2) (2007), 513–530. Zbl 1127.14045, MR 2363430, 10.1112/jlms/jdm075
Reference: [40] Landsberg, J. M., Weyman, J.: On the ideals and singularities of secant varieties of Segre varieties.Bull. London Math. Soc. 39 (4) (2007), 685–697. Zbl 1130.14041, MR 2346950, 10.1112/blms/bdm049
Reference: [41] LeBrun, C., Salamon, S.: Strong rigidity of positive quaternion-Kahler manifolds.Invent. Math. 118 (1994), 109–132. MR 1288469, 10.1007/BF01231528
Reference: [42] Loday, P.: Algebraic operads, Koszul duality and generalized bialgebras.Srní lectures, 2008.
Reference: [43] Robles, C.: Rigidity of the adjoint variety of $\mathfrak{sl}_n$.preprint math.DG/0608471.
Reference: [44] Sasaki, T., Yamaguchi, K., Yoshida, M.: On the rigidity of differential systems modelled on Hermitian symmetric spaces and disproofs of a conjecture concerning modular interpretations of configuration spaces.CR-geometry and overdetermined systems (Osaka, 1994), Adv. Stud. Pure Math. 25, 318-354 (1997), 1997. Zbl 0908.17013, MR 1476250
Reference: [45] Se-Ashi, Y.: On differential invariants of integrable finite type linear differential equations.Hokkaido Math. J. 17 (2) (1988), 151–195. Zbl 0664.34018, MR 0945853
Reference: [46] Vogel, P.: The universal Lie algebra.preprint http://people.math.jussieu.fr/$\tilde{\ }$vogel/.
Reference: [47] Yamaguchi, K.: Differential systems associated with simple graded Lie algebras.Progress in differential geometry, Adv. Stud. Pure Math. 22, 1993. Zbl 0812.17018, MR 1274961
Reference: [48] Yang, D.: Involutive hyperbolic differential systems.Mem. Amer. Math. Soc. 68 (370) (1987), xii+93 pp. Zbl 0639.35057, MR 0897707
.

Files

Files Size Format View
ArchMathRetro_044-2008-5_8.pdf 724.9Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo