Infinitesimal extensions of rank two vector bundles on submanifolds of small codimension

Abstract

Let $X$ be a submanifold of dimension $n$ of the complex projective space $\mathbb P^N$ ($n<N$), and let $E$ be a vector bundle of rank two on $X$ . If $n\geq\frac{N+3}{2}\geq 4$ we prove a geometric criterion for the existence of an extension of $E$ to a vector bundle on the first order infinitesimal neighborhood of $X$ in $\mathbb P^N$ in terms of the splitting of the normal bundle sequence of $Y\subset X\subset\mathbb P^N$, where $Y$ is the zero locus of a general section of a high twist of $E$. In the last section we show that the universal quotient vector bundle on the Grassmann variety $\mathbb G(k,m)$ of $k$-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 $\mathbb P^N$ as a vector bundle.