Extensions of Vector Bundles With Application to Noether–lefschetz Theorems

Abstract

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 [Formula: see text] 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.

Remarks on higher-rank ACM bundles on hypersurfaces

“…We refer the interested reader to [14] for more details. Let n 4, and X ⊂ P n+1 be a smooth hypersurface of degree d, with I its ideal sheaf, and f its defining polynomial.…”

“…is the class of the short exact sequence (3), and η E is the (obstruction) class which vanishes if and only if E extends to a vector bundle on X 1 (see §3.3 in [14] for details).…”

“…and hence defines an element of the group Ext 1 X 1 (E, E(−d)). A standard Leray spectral sequence argument (see [14], Proposition 2) yields a 4-term sequence…”

Grothendieck–Lefschetz theorem with base locus

“…(a) (Moishezon [11]) f is a closed immersion, X is smooth and either (Ravindra and Srinivas [16]) f * K X (1) is generated by global sections. (c) (Ravindra and Tripathi [17]) The multiplication map H 0 (X, O X (1)) ⊗ H 0 (X, K X (1)) → H 0 (X, K X (2)) is surjective and H 1 (X, Ω 2X ⊗ π * O X (1)) = 0 for some desingularization π :X → X. (a) Moishezon proves his result with the original method of Lefschetz [8], using monodromy, Hodge theory, Lefschetz pencils and vanishing cycles (see Voisin's book [21]).…”