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

TL;DR

This paper establishes a geometric criterion for extending rank two vector bundles on certain submanifolds of projective space to their first infinitesimal neighborhoods, and shows the universal quotient bundle on Grassmannians does not extend in this way.

Contribution

It provides a new criterion for infinitesimal extension of vector bundles on submanifolds of small codimension and demonstrates non-extendability of universal quotient bundles on Grassmannians.

Findings

Extension criterion based on normal bundle splitting

Universal quotient bundle on Grassmannians does not extend

Results apply to submanifolds with dimension at least (N+3)/2

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.