Infinitely stably extendable vector bundles on projective spaces

TL;DR

This paper characterizes when a vector bundle on projective space can be stably extended to higher-dimensional projective spaces, linking it to the cohomology of a specific type of monad and providing a new effective version of the Babylonian tower theorem.

Contribution

It establishes a precise criterion for stable extension of vector bundles on projective spaces based on monad cohomology, enhancing understanding of their structure and extension properties.

Findings

E extends stably to P^N iff E is cohomology of a three-term free monad

Provides a new effective version of the Babylonian tower theorem

Uses method of Coanda and Trautmann (2006) in proof

Abstract

According to Horrocks (1966), a vector bundle E on the projective n-space extends stably to the projective N-space, N>n, if there exists a vector bundle on the larger space whose restriction to the smaller one is isomorphic to E plus a direct sum of line bundles. We show that E extends stably to the projective N-space for every N>n if and only if E is the cohomology of a free monad (with three terms). The proof uses the method of Coanda and Trautmann (2006). Combining this result with a theorem of Mohan Kumar, Peterson and Rao (2003), we get a new effective version of the Babylonian tower theorem for vector bundles on projective spaces.