Fano bundles and splitting theorems on projective spaces and quadrics

Abstract

The aim of this paper is to describe the structure of Fano bundles in dimension > 4. Introduction. In this paper rank 2 vector bundles E on projective spaces Ψ n and quadrics Q n are investigated which enjoy the additional property that their projectized bundles Ψ(E) are Fano manifolds, i.e. have negative canonical bundles. Such bundles are shortly called Fano bundles. Up to dimension 3 Fano bundles are completely classified by [SW], [SW], [SW"], [SSW]. The aim of this paper is to describe the structure of Fano bundles in dimension > 4. Namely we prove the following MAIN THEOREM. Let E be a rank 2 Fano bundle on Ψ n or Q n , n > 4. Then up to some explicit exceptions on Q 4 and Q$ (see ex. (2.1), (2.2), (2.3)), E splits into a direct sum of line bundles. A rank 2 bundle E on Ψ n is Fano if and only if the "Q-vector bundle" E <g> (det£*)/2 ® &{?ψ) is ample, i.e.

Fano n-folds with nef tangent bundle and Picard number greater than $$n-5$$ n - 5

“…We will denote by ξ the class of the tautological line bundle O P(E ) (1) and by H the pullback of the ample generator of Pic Y .…”

A classification theorem on Fano bundles

“…Given a symplectic form L in C 4 , the subvariety of G(1, 3) parametrizing lines in P 3 = P(C 4 ) that are isotropic with respect to L is denoted by LG (1,3). It is well known that this variety is a linear section of (the Plücker embedding of) G(1, 3), hence isomorphic to Q 3 .…”

Fano Manifolds and Blow-Ups of Low-Dimensional Subvarieties

“…A natural question which arises from the study of Fano manifolds is to investigate -and possibly classify -Fano manifolds which admit an extremal contraction with special features: for example, this has been done in many cases in which the contraction is a projective bundle [1,18,21,22,23,24], a quadric bundle [29] or a scroll [5,16].…”

“…Let X be a Fano manifold of dimension n ≥ 6 and pseudoindex i X ≥ 2, which is the blow-up of another Fano manifold Y along a smooth subvariety B of dimension i X ; assume that X does not admit a fiber type contraction. Then Y G (1,4) and B is a plane of bidegree (0, 1).…”