Lucian Badescu: Infinitesimal extensions of rank two vector bundles on submanifolds of small codimension, p.231-243

Let

be a submanifold of dimension

of the complex projective space $\mathbb P^N$ (

), and let

be a vector bundle of rank two on

. If $n\geq\frac{N+3}{2}\geq 4$ we prove a geometric criterion for the existence of an extension of

to a vector bundle on the first order infinitesimal neighborhood of

in $\mathbb P^N$ in terms of the splitting of the normal bundle sequence of $Y\subset X\subset\mathbb P^N$ , where

is the zero locus of a general section of a high twist of

. In the last section we show that the universal quotient vector bundle on the Grassmann variety $\mathbb G(k,m)$ of

-dimensional linear subspaces of $\mathbb P^m$ , with $m\geq 3$ and $1\leq k\leq m-2$ (i.e. with $\mathbb G(k,m)$ not a projective space), embedded in any projective space $\mathbb P^N$ , does not extend to the first infinitesimal neighborhood of $\mathbb G(k,m)$ in $\bP^N$ as a vector bundle.

Lucian Badescu: Infinitesimal extensions of rank two vector bundles on submanifolds of small codimension, p.231-243

Abstract: