Vector bundles on projective varieties which split along q-ample subvarieties

TL;DR

This paper establishes new criteria for when a vector bundle on a smooth projective variety splits, based on its behavior along q-ample subvarieties, extending previous splitting results to broader contexts.

Contribution

It provides a unified approach to vector bundle splitting criteria using q-ample subvarieties, applicable to various geometric situations and specific varieties like Grassmannians.

Findings

Vector bundles split if restricted to q-ample subvarieties with certain deformation properties.

Splitting criteria apply to zero loci of sections in globally generated bundles.

Splitting on symplectic and orthogonal Grassmannians can be determined from low-dimensional sub-Grassmannians.

Abstract

Let Y be a subvariety of a smooth projective variety X, and V a vector bundle on X. Given that the restriction of V to Y splits into a direct sum of line bundles, we ask whether V splits on X. I answer this question in affirmative if holds: Y is a q-ample subvariety of X (for appropriate q), it admits sufficiently many embedded deformations, and is very general within its own deformation space. The result goes beyond the previously known splitting criteria for vector bundles corresponding to restrictions. It allows to treat in a unified way examples arising in totally different situations. I discuss the particular cases of zero loci of sections in globally generated vector bundles, on one hand, and sources of multiplicative group actions (corresponding to Bialynicki-Birula decompositions), on the other hand. Finally, I elaborate on the symplectic and orthogonal Grassmannians; I prove that the splitting of any vector bundle on them can be read off from the restriction to a low dimensional `sub'-Grassmannian.