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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 $$G(k,m)$$.
##### MSC:
 14M07 Low codimension problems in algebraic geometry 14M10 Complete intersections