Extensions of vector bundles with application to Noether-Lefschetz theorems (Q2859241)

The authors extend Lefschetz theory for line bundles to vector bundles. One of the typical theorem states that if \(Y\) is a smooth projective variety of dimension at least four, \(X\subset Y\) is a smooth hyperplane section and \(E\) a vector bundle on \(X\) satisfying the vanishing condition, \(H^2(X, \mathcal{E}nd E\otimes \mathcal{O}_Y(-kX)_{|X})=0\) for all \(k>0\), then \(E\) extends to a vector bundle on a Zariski open subset of \(Y\) containing \(X\). The proofs breaks up into an infinitesimal Noether-Lefschetz theorem and then a global version. There are some interesting variations in the infinitesimal version in this article.NEWLINENEWLINE Reviewer's remark: There are a few typos, like in the above theorem \(=0\) is missing and similarly in Theorem 4, due to Grothendieck, required hypotheses are missing.