Identifying quadric bundle structures on complex projective varieties

Geometriae Dedicata - Tập 139 - Trang 289-297 - 2008
Carolina Araujo1
1IMPA, Rio de Janeiro, Brazil

Tóm tắt

In this paper we characterize smooth complex projective varieties that admit a quadric bundle structure on some dense open subset in terms of the geometry of certain families of rational curves.

Tài liệu tham khảo

Andreatta M., Wiśniewski J.A.: On manifolds whose tangent bundle contains an ample subbundle. Invent. Math. 146(1), 209–217 (2001) Araujo C.: Rational curves of minimal degree and characterizations of projective spaces. Math. Ann. 335(4), 937–951 (2006) Araujo C., Druel S., Kovács S.: Cohomological characterizations of projective spaces and hyperquadrics. Invent. Math. 174(2), 233–253 (2008) Beltrametti M.C., Ionescu P.: On manifolds swept out by high dimensional quadrics. Math. Z. 260(1), 229–234 (2008) Bonavero L., Casagrande C., Druel S.: On covering and quasi-unsplit families of rational curves. J. Eur. Math. Soc. (JEMS) 9(1), 45–76 (2007) Campana F.: Connexité rationnelle des variétés de Fano. Ann. Sci. École Norm. Sup. (4) 25(5), 539–545 (1992) Cho, K., Miyaoka, Y., Shepherd-Barron, N.I.: Characterizations of projective space and applications to complex symplectic manifolds , Higher dimensional birational geometry (Kyoto, 1997), Adv. Stud. Pure Math., vol. 35, pp. 1–88. Math. Soc. Japan, Tokyo (2002) Debarre O.: Higher-dimensional algebraic geometry. Universitext, Springer–Verlag, New York (2001) Fujita T.: On the structure of polarized varieties with Δ-genera zero. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22, 103–115 (1975) Graber T., Harris J., Starr J.: Families of rationally connected varieties. J. Amer. Math. Soc. 16(1), 57–67 (2003) (electronic) Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math., No. 28 (1966) Hartshorne R.: Algebraic geometry, Graduate texts in mathematics, vol. 52. Springer–Verlag, New York (1977) Hwang J.-M.: Stability of tangent bundles of low-dimensional Fano manifolds with Picard number 1. Math. Ann. 312(4), 599–606 (1998) Hwang, J.-M.: Geometry of minimal rational curves on Fano manifolds, School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), ICTP Lect. Notes, vol. 6, Abdus Salam Int. Cent. Theoret. Phys., pp. 335–393. Trieste (2001) Hwang J.-M.: Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1. Ann. Inst. Fourier (Grenoble) 57(3), 815–823 (2007) Hwang J.-M., Mok N.: Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation. Invent. Math. 131(2), 393–418 (1998) Hwang J.-M., Mok N.: Birationality of the tangent map for minimal rational curves. Asian J. Math. 8(1), 51–64 (2004) Kebekus S.: Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron, Complex geometry (Göttingen, 2000), pp. 147–155. Springer, Berlin (2002) Kebekus S.: Families of singular rational curves. J. Algebraic Geom. 11(2), 245–256 (2002) Kollár J.: Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer–Verlag, Berlin (1996) Mok, N.: Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents, pre-print 09 of IMR Preprint Series (2005) Mori S.: Projective manifolds with ample tangent bundles. Ann. Math. (2) 110(3), 593–606 (1979)