Splitting Criteria for Vector Bundles Induced by Restrictions to Divisors

A Few Splitting Criteria for Vector Bundles

We prove a few splitting criteria for vector bundles on a quadric hypersurface and Grassmannians. We give also some cohomological splitting conditions for rank 2 bundles on multiprojective spaces. The tools are monads and a Beilinson’s type spectral sequence generalized by Costa and Miró-Roig.

Splitting Criteria for Vector Bundles on Higher Dimensional Varieties

We generalize Horrocks’ criterion for the splitting of vector bundles on projective space. We establish an analogous splitting criterion for vector bundles on a class of smooth complex projective varieties of dimension ≥ 4, over which every extension of line bundles splits.

Normalized Tautological Divisors of Semi-stable Vector Bundles

A Splitting Criterion for Vector Bundles on Higher Dimensional Varieties

We generalize Horrocks’ criterion for the splitting of vector bundles on projective space. We establish an analogous splitting criterion for vector bundles on arbitrary smooth complex projective varieties of dimension ≥ 4, which asserts that a vector bundle E on X splits iff its restriction E|Y to an ample smooth codimension 1 subvariety Y ⊂ X splits.

Divisors and Line Bundles

An analytic hypersurface of M is a subset V ⊂ M such that for each point x ∈ V there exists an open set Ux ⊂ M containing x and a holomorphic function fx defined on Ux such that V ⊂ Ux is the zero-set of fx. Such an fx is called a local defining function for V near x. The quotient of any two local defining functions around x is a non-vanishing holomorphic function around x. An analytic hypersur...