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Differential Geometry of Submanifolds of Projective Space

  • Joseph M. Landsberg
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 144)

Abstract

These are lecture notes on the rigidity of submanifolds of projective space “resembling” compact Hermitian symmetric spaces in their homogeneous embeddings. The results of [16, 20, 29, 18, 19, 10, 31] are surveyed, along with their classical predecessors. The notes include an introduction to moving frames in projective geometry, an exposition of the Hwang-Yamaguchi ridgidity theorem and a new variant of the Hwang-Yamaguchi theorem.

Keywords

Projective Space Fundamental Form Hermitian Symmetric Space Differential Invariant Segre Variety 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    R. Beheshti, Lines on projective hyper surf aces, J. Reine Angew. Math. (2006), 592: 1–21.MATHMathSciNetGoogle Scholar
  2. [2]
    N. Bourbaki, Groupes et algèbres de Lie, Hermann, Paris, 1968, MR0682756.MATHGoogle Scholar
  3. [3]
    E. Cartan, Sur les variétés de courbure constante d’un espace euclidien ou non euclidien, Bull. Soc. Math Prance (1919), 47: 125–160, and (1920), 48: 132-208; see also pp. 321-432 in Oeuvres Complètes Part III, Gauthier-Villars, 1955.MathSciNetGoogle Scholar
  4. [4]
    L. Ein, Varieties with small dual varieties, I, Inventiones Math. (1986), 86: 63–74.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    G. Fubini, Studi relativi all’elemento lineare proiettivo di una ipersuperficie, Rend. Acad. Naz. dei Lincei (1918), 99–106.Google Scholar
  6. [6]
    P.A. Griffiths AND J. Harris, Algebraic Geometry and Local Differential Geometry, Ann. scient. Ec. Norm. Sup. (1979), 12: 355–432, MR0559347.MATHMathSciNetGoogle Scholar
  7. [7]
    J.-M. Hwang, Geometry of minimal rational curves on Fano manoflds, ICTP lecture notes, www.ictp.trieste.it./~pub_off/services.Google Scholar
  8. [8]
    J.-M. Hwang AND N. Mok, Uniruled projective manifolds with irreducible reductive G-structures, J. Reine Angew. Math. (1997), 490: 55–64, MR1468924.MATHMathSciNetGoogle Scholar
  9. [9]
    W.V.D. Hodge AND D. Pedoe, Methods of algebraic geometry, Vol. II, Cambridge University Press, Cambridge (1994), p. 394+.Google Scholar
  10. [10]
    J.-M. Hwang AND K. Yamaguchi, Characterization of Hermitian symmetric spaces by fundamental forms, Duke Math. J. (2003), 120(3): 621–634.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    T. Ivey AND J.M. Landsberg, Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, 61, American Mathematical Society, Providence, RI, 2003, MR2003610.Google Scholar
  12. [12]
    G. Jensen AND E. Musso, Rigidity of hyper surf aces in complex projective space, Ann. scient. Ec. Norm. (1994), 27: 227–248.MATHMathSciNetGoogle Scholar
  13. [13]
    B. Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. of Math. (1961), 74(2): 329–387. MR0142696CrossRefMathSciNetGoogle Scholar
  14. [14]
    J.M. Landsberg, On second fundamental forms of projective varieties, Inventiones Math. (1994), 117: 303–315, MR1273267.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    —, Differential-geometric characterizations of complete intersections, J. Differential Geom. (1996), 44: 32–73, MR1420349.MATHMathSciNetGoogle Scholar
  16. [16]
    —, On the infinitesimal rigidity of homogeneous varieties, Compositio Math. (1999), 118: 189–201, MR1713310.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    —, Algebraic geometry and projective differential geometry, Seoul National University concentrated lecture series 1997, Seoul National University Press, 1999, MR1712467.Google Scholar
  18. [18]
    —, Is a linear space contained in a submanifold? — On the number of derivatives needed to tell, J. reine angew. Math. (1999), 508: 53–60.MATHMathSciNetGoogle Scholar
  19. [19]
    —, Lines on projective varieties, J. Reine Angew. Math. (2003), 562: 1–3, MR2011327MATHMathSciNetGoogle Scholar
  20. [20]
    —, Griffiths-H arris rigidity of compact Hermitian symmetric spaces, J. Differential Geometry (2006), 74: 395–405.MATHMathSciNetGoogle Scholar
  21. [21]
    J.M. Landsberg AND L. Manivel, On the projective geometry of rational homogeneous varieties, Comment. Math. Helv. (2003), 78(1): 65–100, MR1966752.MATHMathSciNetGoogle Scholar
  22. [22]
    —, Classification of simple Lie algebras via projective geometry, Selecta Mathematica (2002), 8: 137–159, MR1890196.MATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    —, Representation theory and projective geometry, Algebraic Transformation Groups and Algebraic Varieties, Ed. V.L. Popov, Encyclopaedia of Mathematical Sciences, Springer 2004, 132: 71–122.MathSciNetGoogle Scholar
  24. [24]
    —, Series of Lie groups, Michigan Math. J. (2004), 52(2): 453–479, MR2069810.MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    —, Triality, exceptional Lie algebras, and Deligne dimension formulas, Adv. Math. (2002), 171: 59–85.MATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    —, The sextonions and E 7 1/2, Adv. Math. (2006), 201(1): 143–179, MR2204753MATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    —, A universal dimension formula for complex simple Lie algebras, Adv. Math. (2006), 201(2): 379–407, MR2211533MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    —, Legendrian varieties, math.AG/0407279. To appear in Asian Math. J.Google Scholar
  29. [29]
    J.M. Landsberg AND C. Robles, Pubin’s theorem in codimension two, preprint math.AG/0509227.Google Scholar
  30. [30]
    E. Mezzetti AND D. Portelli, On threefolds covered by lines, Abh. Math. Sem. Univ. Hamburg (2000), 70: 211–238, MR1809546.MATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    C. Robles, Rigidity of the adjoint variety of sin, preprint math.DG/0608471.Google Scholar
  32. [32]
    E. Rogora, Varieties with many lines, Manuscripta Math. (1994), 82(2): 207–226.MATHCrossRefMathSciNetGoogle Scholar
  33. [33]
    T. Sasaki, K. Yamaguchi, AND M. Yoshida, On the rigidity of differential systems modelled on Hermitian symmetric spaces and disproofs of a conjecture concerning modular interpretations of configuration spaces, in CR-geometry and overdetermined systems (Osaka, 1994), 318–354, Adv. Stud. Pure Math., 25, Math. Soc. Japan, Tokyo, 1997.Google Scholar
  34. [34]
    Y. Se-Ashi, On differential invariants of integrable finite type linear differential equations, Hokkaido Math. J. (1988), 17(2): 151–195. MR0945853MATHMathSciNetGoogle Scholar
  35. [35]
    B. Segre, Bertini forms and Hessian matrices, J. London Math. Soc. (1951), 26: 164–176.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Joseph M. Landsberg
    • 1
  1. 1.Department of MathematicsTexas A & M UniversityCollege Station

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