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 $n\ge \frac{N+3}{2}\ge 4$ 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 $m\ge 3$ and $1\le k\le m-2$ (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.

arXiv commentTo appear in Bulletin Mathematique de la Societe des Sciences Mathematiques de la Romanie