Geometric structures on uniruled projective manifolds defined by their varieties of minimal rational tangents
Géométrie différentielle, physique mathématique, mathématiques et société (II) - Volume en l'honneur de Jean-Pierre Bourguignon, Astérisque no. 322  (2008), p. 151-205
@incollection{AST_2008__322__151_0,
     author = {Mok, Ngaiming},
     title = {Geometric structures on uniruled projective manifolds defined by their varieties of minimal rational tangents},
     booktitle = {G\'eom\'etrie diff\'erentielle, physique math\'ematique, math\'ematiques et soci\'et\'e (II) - Volume en l'honneur de Jean-Pierre Bourguignon},
     editor = {Hijazi Oussama},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {322},
     year = {2008},
     pages = {151-205},
     zbl = {1182.14043},
     mrnumber = {2521656},
     language = {en},
     url = {http://www.numdam.org/item/AST_2008__322__151_0}
}
Mok, Ngaiming. Geometric structures on uniruled projective manifolds defined by their varieties of minimal rational tangents, in Géométrie différentielle, physique mathématique, mathématiques et société (II) - Volume en l'honneur de Jean-Pierre Bourguignon, Astérisque, no. 322 (2008), pp. 151-205. http://www.numdam.org/item/AST_2008__322__151_0/

[1] W. M. Boothby - "Homogeneous complex contact manifolds", in Proc. Sympos. Pure Math., Vol. III, Amer. Math. Soc, 1961, p. 144-154. | Article | MR 124863 | Zbl 0103.38702

[2] F. Campana & T. Peternell - "Projective manifolds whose tangent bundles are numerically effective", Math. Ann. 289 (1991), p. 169-187. | Article | MR 1087244 | Zbl 0729.14032

[3] K. Cho, Y. Miyaoka & N. I. Shepherd-Barron - "Characterizations of projective space and applications to complex symplectic manifolds", in Higher dimensional birational geometry (Kyoto, 1997), Adv. Stud. Pure Math., vol. 35, Math. Soc. Japan, 2002, p. 1-88. | MR 1929792 | Zbl 1063.14065

[4] A. Grothendieck - "Sur la classification des fibrés holomorphes sur la sphère de Riemann", Amer. J. Math. 79 (1957), p. 121-138. | Article | MR 87176 | Zbl 0079.17001

[5] V. Guillemin - "The integrability problem for G-structures", Trans. Amer. Math. Soc. 116 (1965), p. 544-560. | MR 203626 | Zbl 0178.55702

[6] J. Hong - "Fano manifolds with geometric structures modeled after homogeneous contact manifolds", Internat. J. Math. 11 (2000), p. 1203-1230. | Article | MR 1809309 | Zbl 1077.14547

[7] J. Hong, "Rigidity of smooth Schubert varieties in Hermitian symmetric spaces", Trans. Amer. Math. Soc. 359 (2007), p. 2361-2381. | Article | MR 2276624 | Zbl 1126.14010

[8] J. Hong & J.-M. Hwang - "Characterization of the rational homogeneous manifold associated to a long simple root by its variety of minimal rational tangents", preprint. | MR 2409558 | Zbl 1186.14044

[9] J. Hong & N. Mok - "Non-equidimensional Cartan-Fubini extension of holomorphic maps respecting varieties of minimal rational tangents", in preparation.

[10] J.-M. Hwang - "Rigidity of homogeneous contact manifolds under Fano deformation", J. reine angew. Math. 486 (1997), p. 153-163. | MR 1450754 | Zbl 0876.53030

[11] J.-M. Hwang, "Geometry of minimal rational curves on Fano manifolds", in School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), ICTP Lect. Notes, vol. 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, p. 335-393. | MR 1919462 | Zbl 1086.14506

[12] J.-M. Hwang, "Rigidity of rational homogeneous spaces", in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zurich, 2006, p. 613-626. | MR 2275613 | Zbl 1096.14035

[13] J.-M. Hwang, "Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1", Ann. Inst. Fourier (Grenoble) 57 (2007), p. 815-823. | Article | Numdam | MR 2336831 | Zbl 1126.32011

[14] J.-M. Hwang & N. Mok - "Uniruled projective manifolds with irreducible reductive G-structures", J. reine angew. Math. 490 (1997), p. 55-64. | MR 1468924 | Zbl 0882.22007

[15] J.-M. Hwang & N. Mok, "Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation", Invent. Math. 131 (1998), p. 393-418. | Article | MR 1608587 | Zbl 0902.32014

[16] J.-M. Hwang & N. Mok, "Holomorphic maps from rational homogeneous spaces of Picard number 1 onto projective manifolds", Invent. Math. 136 (1999), p. 209-231. | Article | MR 1681093 | Zbl 0963.32007

[17] J.-M. Hwang & N. Mok, "Varieties of minimal rational tangents on uniruled projective manifolds", in Several complex variables (Berkeley, CA, 1995-1996), Math. Sci. Res. Inst. Publ., vol. 37, Cambridge Univ. Press, 1999, p. 351-389. | MR 1748609 | Zbl 0978.53118

[18] J.-M. Hwang & N. Mok, "Cartan-Fubini type extension of holomorphic maps for Fano manifolds of Picard number 1", J. Math. Pures Appl. 80 (2001), p. 563-575. | Article | MR 1842290 | Zbl 1033.32013

[19] J.-M. Hwang & N. Mok, "Deformation rigidity of the rational homogeneous space associated to a long simple root", Ann. Sci. École Norm. Sup. 35 (2002), p. 173-184. | Article | Numdam | MR 1914930 | Zbl 1008.32012

[20] J.-M. Hwang & N. Mok, "Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles", J. Algebraic Geom. 12 (2003), p. 627-651. | Article | MR 1993759 | Zbl 1038.14018

[21] J.-M. Hwang & N. Mok, "Birationality of the tangent map for minimal rational curves", Asian J. Math. 8 (2004), p. 51-63. | Article | MR 2128297 | Zbl 1072.14015

[22] J.-M. Hwang & N. Mok, "Deformation rigidity of the 20-dimensional F 4 -homogeneous space associated to a short root", in Algebraic transformation groups and algebraic varieties, Encyclopaedia Math. Sci., vol. 132, Springer, 2004, p. 37-58. | Article | MR 2090669 | Zbl 1071.22012

[23] J.-M. Hwang & N. Mok, "Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler deformation", Invent. Math. 160 (2005), p. 591-645. | Article | MR 2178704 | Zbl 1071.32022

[24] J.-M. Hwang & S. Ramanan - "Hecke curves and Hitchin discriminant", Ann. Sci. École Norm. Sup. 37 (2004), p. 801-817. | Article | Numdam | MR 2103475 | Zbl 1073.14046

[25] S. Kebekus - "Lines on contact manifolds", J. reine angew. Math. 539 (2001), p. 167-177. | MR 1863858 | Zbl 0983.53031

[26] S. Kebekus, "Families of singular rational curves", J. Algebraic Geom. 11 (2002), p. 245-256. | Article | MR 1874114 | Zbl 1054.14035

[27] S. Kebekus, T. Peternell, A. J. Sommese & J. A. Wiśniewski - "Projective contact manifolds", Invent. Math. 142 (2000), p. 1-15. | Article | MR 1784795 | Zbl 0994.53024

[28] S. Kobayashi & T. Ochiai - "Holomorphic structures modeled after compact Hermitian symmetric spaces", in Manifolds and Lie groups (Notre Dame, Ind., 1980), Progr. Math., vol. 14, Birkhäuser, 1981, p. 207-222. | Article | MR 642859 | Zbl 0498.53038

[29] J. Kollár - Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 32, Springer, 1996. | MR 1440180 | Zbl 0877.14012

[30] J. Kollár, Y. Miyaoka & S. Mori - "Rational connectedness and boundedness of Fano manifolds", J. Differential Geom. 36 (1992), p. 765-779. | Article | MR 1189503 | Zbl 0759.14032

[31] C.-H. Lau - "Holomorphic maps from rational homogeneous manifolds onto projective manifolds", to appear in J. Alg. Geom. | MR 2475814 | Zbl 1160.32018

[32] R. Lazarsfeld - "Some applications of the theory of positive vector bundles", in Complete intersections (Acireale, 1983), Lecture Notes in Math., vol. 1092, Springer, 1984, p. 29-61. | Article | MR 775876 | Zbl 0547.14009

[33] C. Lebrun - "A rigidity theorem for quaternionic-Kähler manifolds", Proc. Amer. Math. Soc. 103 (1988), p. 1205-1208. | MR 955010 | Zbl 0655.53056

[34] C. Lebrun, "Fano manifolds, contact structures, and quaternionic geometry", Internat. J. Math. 6 (1995), p. 419-437. | Article | MR 1327157 | Zbl 0835.53055

[35] Y. Matsushima & A. Morimoto - "Sur certains espaces fibrés holomorphes sur une variété de Stein", Bull Soc. Math. France 88 (1960), p. 137-155. | Article | Numdam | MR 123739 | Zbl 0094.28104

[36] Y. Miyaoka - "Geometry of rational curves on varieties", in Geometry of higher-dimensional algebraic varieties, Birkhäuser, 1997, p. 1-127. | Zbl 0872.14007

[37] Y. Miyaoka, "Numerical characterisations of hyperquadrics", in Complex analysis in several variables - Memorial Conference of Kiyoshi Oka's Centennial Birthday, Adv. Stud. Pure Math., vol. 42, Math. Soc. Japan, 2004, p. 209-235. | Article | MR 2087053 | Zbl 1072.14528

[38] Y. Miyaoka & S. Mori - "A numerical criterion for uniruledness", Ann. of Math. 124 (1986), p. 65-69. | Article | MR 847952 | Zbl 0606.14030

[39] N. Mok - "The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature", J. Differential Geom. 27 (1988), p. 179-214. | Article | MR 925119 | Zbl 0642.53071

[40] N. Mok, "G-structures on irreducible Hermitian symmetric spaces of rank 2 and deformation rigidity", in Complex geometric analysis in Pohang (1997), Contemp. Math., vol. 222, Amer. Math. Soc, 1999, p. 81-107. | Article | MR 1653044 | Zbl 0929.53022

[41] N. Mok, "On Fano manifolds with nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents", Trans. Amer. Math. Soc. 354 (2002), p. 2639-2658. | Article | MR 1895197 | Zbl 0998.32013

[42] N. Mok, "Characterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents", Sci. China Ser. A 51 (2008), p. 660-684. | Article | MR 2395412 | Zbl 1165.32008

[43] N. Mok, "Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents", AMS/IP Studies in Advanced Mathematics 48 (2008), p. 41-61. | MR 2409622 | Zbl 1182.14042

[44] N. Mok & I. H. Tsai - "Rigidity of convex realizations of irreducible bounded symmetric domains of rank 2", J. reine angew. Math. 431 (1992), p. 91-122. | MR 1179334 | Zbl 0765.32017

[45] S. Mori - "Projective manifolds with ample tangent bundles", Ann. of Math. 110 (1979), p. 593-606. | Article | MR 554387 | Zbl 0423.14006

[46] Y. A. Neretin - "Conformal geometry of symmetric spaces, and generalized linear-fractional Kreĭn-Smuľyan mappings", Mat. Sb. 190 (1999), p. 93-122. | MR 1701002 | Zbl 0967.53036

[47] T. Ochiai - "Geometry associated with semisimple flat homogeneous spaces", Trans. Amer. Math. Soc. 152 (1970), p. 159-193. | Article | MR 284936 | Zbl 0205.26004

[48] J-P. Serre - Algèbres de Lie semi-simples complexes, W. A. Benjamin, inc., New York-Amsterdam, 1966. | MR 215886 | Zbl 0144.02105

[49] I. M. Singer & S. Sternberg - "The infinite groups of Lie and Cartan. I. The transitive groups", J. Analyse Math. 15 (1965), p. 1-114. | Article | MR 217822 | Zbl 0277.58008

[50] Y. T. Siu & S. T. Yau - "Compact Kähler manifolds of positive bisectional curvature", Invent. Math. 59 (1980), p. 189-204. | Article | MR 577360 | Zbl 0442.53056

[51] K. Yamaguchi - "Differential systems associated with simple graded Lie algebras", in Progress in differential geometry, Adv. Stud. Pure Math., vol. 22, Math. Soc. Japan, 1993, p. 413-494. | Article | MR 1274961 | Zbl 0812.17018

[52] F. L. Zak - Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, vol. 127, Amer. Math. Soc., 1993. | MR 1234494 | Zbl 0795.14018

[53] F. Zheng - "On semi-positive threefolds", Ph.D. Thesis, Harvard University, 1990.