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Infinitesimal extensions of rank two vector bundles on submanifolds of small codimension. (English) Zbl 1399.14019
In this paper the author studies the following question: Let \(E\) be a vector bundle on a submanifold \(X\) of dimension \(n\) of the complex projective manifold \(P\) of dimension \(N\) and \(X(i)\) the ith infinitesimal neighborhood of \(X\) in \(P\). He first proves a necessary and sufficient geometric criterion for extending a rank two bundle \(E\) on \(X\) to a vector bundle on the first infinitesimal neighborhood \(X(1)\) of \(X\) in \(P\) without assuming a Grothendieck vanishing condition [A. Grothendieck (ed.) and M. Raynaud, Lect. Notes Math. 224, 447 p. (1971; Zbl 0234.14002)]. His criterion involves the splitting of a normal bundle sequence and the use of a generalized form of the Hartshorne-Serre correspondence, whose proof is given, written jointly with E. Arrondo [Rev. Mat. Complut. 20, No. 2, 423–443 (2007; Zbl 1133.14046)]. In addition the author proves that the universal quotient vector bundle of the Grassmann variety \(G(k,m)\) of linear subspaces of dimension \(k\) in the \(m\) dimensional projective space never extends as a vector bundle to the first infinitesimal neighborhood of \(G(k,m)\) with respect to any projective embedding of \(G(k,m)\).
14M07 Low codimension problems in algebraic geometry
14M10 Complete intersections