EXTENSIONS OF VECTOR BUNDLES WITH APPLICATION TO NOETHER-LEFSCHETZ THEOREMS G. V. RAVINDRA AND AMIT TRIPATHI Abstract. Given a smooth, projective variety Y over an algebraically closed field of charac- teristic zero, and a smooth, ample hyperplane section X ⊂ Y , we study the question of when a bundle E on X, extends to a bundle E 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. 1. Introduction We work over an algebraically closed field of chacteristic zero, which we denote by k. One of the most fundamental results in algebraic geometry are the Lefschetz theorems which state that if Y is a smooth, projective variety and X ⊂ Y is a smooth member of an ample linear system, then the Picard groups of Y and X are isomorphic provided dim X ≥ 3; when dim X = 2, the same is true, if in addition, we assume that X is a very general member of a sufficiently ample linear system. These theorems imply in particular, that any line bundle on X extends to a line bundle on Y . From this point of view, one may ask if there are analogous results for higher rank bundles. Let L be an ample line bundle on a smooth projective variety Y , and X be a smooth member of the associated linear system | H 0 (Y,L)|. For k ≥ 0, let X k denote the k-th order thickening of X in Y , so that X 0 = X . The obstruction for a bundle E on X k−1 to lift to a bundle on X k is a class η E ∈ H 2 (X, E nd E ⊗O Y (−kX ) |X ) (see §2). Clearly, the vanishing of these classes is a necessary condition for E to extend to the ambient variety Y . The fact that these classes depend on the bundle E is one of the main points of departure when we study extension questions for higher rank bundles; when E is a line bundle, E nd E ∼ = O X , and so the obstruction classes do not depend on the bundle per se. Consequently, one cannot hope to get a uniform result for all bundles of any given rank. Another noteworthy point when studying higher rank bundles is that even if these obstruction classes vanish, in most cases the bundle extends only as a reflexive sheaf on Y , and would need to satisfy additional conditions in order for the extension to be a bundle. Consider for example, the inclusion P n  → P n+1 , and let π x : P n+1 \{x}→ P n denote the projection map for x ∈ P n+1 \ P n . Since the composition P n  → P n+1 \{x}→ P n is the identity map, the pull-back bundle π ∗ x E for any bundle E on P n is an extension of E on the variety P n+1 \{x}. Even if π ∗ x E extends to a bundle on P n+1 , note that there exists an N>n such that E does not extend to a bundle on P N . For if this were to be so, then by the Babylonian theorems (see [1] for the rank 2 case, and [18] for arbitrary rank), E would have to be a sum of line bundles. The following result (see also [6]) summarises the discussion above. Date : March 28, 2013. 1