Rank two bundles on P^n with isolated cohomology

TL;DR

This paper investigates the cohomological properties of rank two vector bundles on projective space, establishing conditions under which such bundles are decomposable based on their isolated cohomology modules.

Contribution

It characterizes when rank two bundles with specific isolated cohomology vanishings are necessarily decomposable, extending understanding of their structure on projective spaces.

Findings

On P^8, bundles with vanishing H^3 and H^4 are decomposable.

For n ≥ 4k, no indecomposable bundles have all but four specific cohomology modules vanishing.

Provides criteria linking cohomology vanishing patterns to bundle decomposability.

Abstract

The purpose of this paper is to study minimal monads associated to a rank two vector bundle $\mathcal E$ on $\mathbb P^n$. In particular, we study situations where $\mathcal E$ has $H^i_*(\mathcal E) =0$ for $1<i<n-1$, except for one pair of values $(k,n-k)$. We show that on $\mathbb P^8,$ if $H^3_*(\mathcal E)=H^4_*(\mathcal E)=0$, then $\mathcal E$ must be decomposable. More generally, we show that for $n\geq 4k$, there is no indecomposable bundle $\mathcal E$ for which all intermediate cohomology modules except for $H^1_*, H^k_*, H^{n-k}_*, H^{n-1}_*$ are zero.