Monads and Rank Three Vector Bundles on Quadrics

In this paper we give the classification of rank 3 vector bundles without ”inner” cohomology on a quadric hypersurface Qn (n > 3) by studying the associated monads. Introduction A monad on P or, more generally, on a projective variety X, is a complex of three vector bundles 0 → A α −→ B β −→ C → 0 such that α is injective as a map of vector bundles and β is surjective. Monads have been studied by Horrocks, who proved (see [Ho] or [BH]) that every vector bundle on P is the homology of a suitable minimal monad. Throughout the paper we often use the Horrocks correspondence between a bundle E on P (n ≥ 3) and the corresponding minimal monad 0 → A α −→ B β −→ C → 0, where A and C are sums of line bundles and B satisfies: 1. H1 ∗ (B) = H n−1 ∗ (B) = 0 2. H i ∗(B) = H i ∗(E) ∀i, 1 < i < n− 1. where H i ∗(B) is defined as ⊕k∈(Z)H (P, B(k)). This correspondence holds also on a projective variety X (dimX ≥ 3) if we fix a very ample line bundle OX(1). Indeed the proof of the result in ([BH] proposition 3) can be easily extended to X (see [Ml])). Rao, Mohan Kumar and Peterson have successfully used this tool to investigate the intermediate cohomology modules of a vector bundle on P and give cohomological splitting conditions (see [KPR1]). This theorem makes a strong use of monads and of Horrocks’ splitting criterion which states the following: Let E be a vector bundle of rank r on P, n ≥ 2 then E splits if and only if it does not have intermediate cohomology (i.e. H1 ∗ (E) = ... = H n−1 ∗ (E) = 0). Mathematics Subject Classification 2000: 14F05, 14J60.