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    Given a smooth, projective variety Y over an algebraically closed field of characteristic zero, and a smooth, ample hyperplane section X ⊂ Y, we study the question of when a bundle E on X, extends to a bundle on a Zariski open set U ⊂ Y containing X. The main ingredients used are explicit descriptions of various obstruction classes in the deformation theory of bundles, together with Grothendieck–Lefschetz theory. As a consequence, we prove a Noether–Lefschetz theorem for higher rank bundles, which recovers and unifies the Noether–Lefschetz theorems of Joshi and Ravindra–Srinivas.

    AMSC: 14B10, 14C22, 14J60


    • W.   Barth and A.   Van de Ven , Invent. Math.   25 , 91 ( 1974 ) . Web of ScienceGoogle Scholar
    • J. Brevik and S. Nollet, Int. Math. Res. Not. 2011(6), 1220 (2011). Web of ScienceGoogle Scholar
    • J. Carlsonet al., Compositio Math. 50(3), 109 (1983). Web of ScienceGoogle Scholar
    • P.   Deligne and N.   Katz , Séminaire de Géométrie Algébrique du Bois-Marie — 1967–1969. Groupes de Monodromie en Géométrie Algébrique. II , Lecture Notes in Mathematics   340 ( Springer-Verlag , 1973 ) . Google Scholar
    • G. Ellingsrud and C. Peskine, Anneau de Gorenstein associé à un fibré inversible sur une surface de l'espace et lieu de Noether–Lefschetz, Proc. Indo-French Conference on Geometry (Hindustan Book Agency, Delhi, 1993) pp. 29–42. Google Scholar
    • T. Fujita, J. Math. Soc. Japan 33(3), 405 (1981). Web of ScienceGoogle Scholar
    • M. L. Green, J. Differential Geom. 27(1), 155 (1988). Web of ScienceGoogle Scholar
    • M. L. Green, J. Differential Geom. 29(2), 295 (1989). Web of ScienceGoogle Scholar
    • A.   Grothendieck , Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux , Advanced Studies in Pure Mathematics   2 ( North-Holland , Amsterdam , 1968 ) . Google Scholar
    • R.   Hartshorne , Deformation Theory , Graduate Texts in Mathematics ( Springer , 2010 ) . Google Scholar
    • D. Huybrechts and R. P. Thomas, Math. Ann. 346(3), 545 (2010). Web of ScienceGoogle Scholar
    • K. Joshi, J. Algebraic Geom. 4(1), 105 (1995). Google Scholar
    • A. F. Lopez, Noether–Lefschetz Theory and the Picard Group of Projective Surfaces, Memoirs of the American Mathematical Society 89 (American Mathematical Society, Providence, RI, 1991) p. x+100. Google Scholar
    • N. Mohan Kumar, A. P. Rao and G. V. Ravindra, Hodge style Chern classes for vector bundles on schemes in characteristic zero, Expository notes . Google Scholar
    • N. Mohan Kumar and V. Srinivas, The Noether–Lefschetz theorem, unpublished notes . Google Scholar
    • G. V. Ravindra and V. Srinivas, J. Algebraic Geom. 15(3), 563 (2006). Web of ScienceGoogle Scholar
    • G. V. Ravindra and V. Srinivas, J. Algebra 322(9), 3373 (2009). Web of ScienceGoogle Scholar
    • E.-I. Sato, J. Math. Kyoto Univ. 17(1), 127 (1977). Google Scholar
    • C. Voisin, Math. Ann. 280(4), 605 (1988). Web of ScienceGoogle Scholar
    • C. Voisin, Comment. Math. Helv. 64(4), 515 (1989). Web of ScienceGoogle Scholar
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