Previous |  Up |  Next

Article

Keywords:
projective rigidity; exterior differential systems; Lie algebra cohomology; homogeneous varieties
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.
References:
[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. DOI 10.1073/pnas.78.8.4657 | MR 0627257 | Zbl 0468.53040
[2] Bourbaki, N.: Groupes et algèbres de Lie. Hermann, Paris, 1968. MR 0240238 | Zbl 0186.33001
[3] Brion, M.: Spherical varieties. Proceedings of the International Congress of Mathematicians, Zürich 1994, vol. 1, 2, Birkhäuser, Basel, 1995, pp. 753–760. MR 1403975 | Zbl 0862.14031
[4] Bryant, R.: Metrics with exceptional holonomy. Ann. of Math. (2) 126 (3) (1987), 525–576. DOI 10.2307/1971360 | MR 0916718 | Zbl 0637.53042
[5] Bryant, R. L.: Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spaces. Princeton University Press, AM-153, 2005.
[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. DOI 10.1007/978-1-4613-9714-4_5 | MR 1083148 | Zbl 0726.58002
[7] Čap, A.: Lie algebra cohomology and overdetermined systems. preprint.
[8] Čap, A., Schichl, H.: Parabolic geometries and canonical Cartan connections. Hokkaido Math. J. 29 (3) (2000), 453–505. MR 1795487
[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.
[10] Cartan, E.: Sur les variétés a connexion projective. Bull. Soc. Math. France 52 (1924), 205–241. MR 1504846
[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. DOI 10.1007/BFb0096224 | MR 0655419 | Zbl 0477.53056
[12] Deligne, P.: La série exceptionnelle des groupes de Lie. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), 321–326. MR 1378507
[13] Deligne, P., de Man, R.: The exceptional series of Lie groups. C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), 577–582. MR 1411045 | Zbl 0910.22009
[14] Fubini, G.: Studi relativi all’elemento lineare proiettivo di una ipersuperficie. Rend. Accad. Naz. dei Lincei (1918), 99–106.
[15] Griffiths, P. A., Harris, J.: Algebraic geometry and local differential geometry. Ann. Sci. École Norm. Sup. (4) 12 (1979), 355–432. MR 0559347 | Zbl 0426.14019
[16] Harvey, R., Lawson, H. B.: Calibrated geometries. Acta Math. 148 (1982), 47–157. DOI 10.1007/BF02392726 | MR 0666108 | Zbl 0584.53021
[17] Hilgert, J.: Multiplicity free branching laws for unitary representations. Srní lectures, 2008.
[18] Hong, J.: Rigidity of singular Schubert varieties in ${\mathrm{G}r}(m,n)$. J. Differential Geom. 71 (1) (2005), 1–22. MR 2191767
[19] Hong, J.: Rigidity of smooth Schubert varieties in Hermitian symmetric spaces. Trans. Amer. Math. Soc. 359 (5) (2007), 2361–2381. DOI 10.1090/S0002-9947-06-04041-4 | MR 2276624 | Zbl 1126.14010
[20] Hwang, J. M., Yamaguchi, K.: Characterization of Hermitian symmetric spaces by fundamental forms. Duke Math. J. 120 (3) (2003), 621–634. DOI 10.1215/S0012-7094-03-12035-9 | MR 2030098 | Zbl 1053.32012
[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. MR 2003610 | Zbl 1105.53001
[22] Kebekus, S., Peternell, T., Sommese, A., Wiśniewski, J.: Projective contact manifolds. Invent. Math. 142 (1) (2000), 1–15. DOI 10.1007/PL00005791 | MR 1784795
[23] Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. of Math. (2) 74 (1961), 329–387. DOI 10.2307/1970237 | MR 0142696 | Zbl 0134.03501
[24] Landsberg, J., Robles, C.: Fubini-Griffiths-Harris rigidity and Lie algebra cohomology. preprint arXiv:0707.3410.
[25] Landsberg, J. M.: Differential-geometric characterizations of complete intersections. J. Differential Geom. 44 (1996), 32–73. MR 1420349 | Zbl 0873.53007
[26] Landsberg, J. M.: On the infinitesimal rigidity of homogeneous varieties. Compositio Math. 118 (1999), 189–201. DOI 10.1023/A:1017161326705 | MR 1713310 | Zbl 0981.53039
[27] Landsberg, J. M.: Griffiths-Harris rigidity of compact Hermitian symmetric spaces. J. Differential Geom. 74 (3) (2006), 395–405. MR 2269783 | Zbl 1107.53036
[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. MR 2384708 | Zbl 1148.53009
[29] Landsberg, J. M.: Geometry and the complexity of matrix multiplication. Bull. Amer. Math. Soc., New Ser. 45 (2) (2008), 247–284. DOI 10.1090/S0273-0979-08-01176-2 | MR 2383305 | Zbl 1145.68054
[30] Landsberg, J. M., Manivel, L.: The projective geometry of Freudenthal’s magic square. J. Algebra 239 (2) (2001), 477–512. DOI 10.1006/jabr.2000.8697 | MR 1832903 | Zbl 1064.14053
[31] Landsberg, J. M., Manivel, L.: Construction and classification of complex simple Lie algebras via projective geometry. Selecta Math. 8 (2002), 137–159. DOI 10.1007/s00029-002-8103-5 | MR 1890196 | Zbl 1073.14551
[32] Landsberg, J. M., Manivel, L.: Triality, exceptional Lie algebras, and Deligne dimension formulas. Adv. Math. 171 (2002), 59–85. DOI 10.1006/aima.2002.2071 | MR 1933384 | Zbl 1035.17016
[33] Landsberg, J. M., Manivel, L.: On the projective geometry of rational homogeneous varieties. Comment. Math. Helv. 78 (1) (2003), 65–100. MR 1966752 | Zbl 1048.14032
[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. MR 2090671 | Zbl 1145.14316
[35] Landsberg, J. M., Manivel, L.: Series of Lie groups. Michigan Math. J. 52 (2) (2004), 453–479. DOI 10.1307/mmj/1091112085 | MR 2069810 | Zbl 1165.17302
[36] Landsberg, J. M., Manivel, L.: A universal dimension formula for complex simple Lie algebras. Adv. Math. 201 (2) (2006), 379–407. DOI 10.1016/j.aim.2005.02.007 | MR 2211533 | Zbl 1151.17003
[37] Landsberg, J. M., Manivel, L.: The sextonions and $E_{7\frac{1}{2}}$. Adv. Math. 201 (1) (2006), 143–179. MR 2204753
[38] Landsberg, J. M., Manivel, L.: Legendrian varieties. Asian Math. J. 11 (3) (2007), 341–360. DOI 10.4310/AJM.2007.v11.n3.a1 | MR 2372722 | Zbl 1136.14024
[39] Landsberg, J. M., Weyman, J.: On tangential varieties of rational homogeneous varieties. J. London Math. Soc. (2) 76 (2) (2007), 513–530. DOI 10.1112/jlms/jdm075 | MR 2363430 | Zbl 1127.14045
[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. DOI 10.1112/blms/bdm049 | MR 2346950 | Zbl 1130.14041
[41] LeBrun, C., Salamon, S.: Strong rigidity of positive quaternion-Kahler manifolds. Invent. Math. 118 (1994), 109–132. DOI 10.1007/BF01231528 | MR 1288469
[42] Loday, P.: Algebraic operads, Koszul duality and generalized bialgebras. Srní lectures, 2008.
[43] Robles, C.: Rigidity of the adjoint variety of $\mathfrak{sl}_n$. preprint math.DG/0608471.
[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. MR 1476250 | Zbl 0908.17013
[45] Se-Ashi, Y.: On differential invariants of integrable finite type linear differential equations. Hokkaido Math. J. 17 (2) (1988), 151–195. MR 0945853 | Zbl 0664.34018
[46] Vogel, P.: The universal Lie algebra. preprint http://people.math.jussieu.fr/$\tilde{\ }$vogel/.
[47] Yamaguchi, K.: Differential systems associated with simple graded Lie algebras. Progress in differential geometry, Adv. Stud. Pure Math. 22, 1993. MR 1274961 | Zbl 0812.17018
[48] Yang, D.: Involutive hyperbolic differential systems. Mem. Amer. Math. Soc. 68 (370) (1987), xii+93 pp. MR 0897707 | Zbl 0639.35057
Partner of
EuDML logo