Vector Bundles on Products of Varieties with $n$-blocks Collections

TL;DR

This paper studies vector bundles on products of varieties with n-block collections, providing cohomological splitting conditions, a characterization method, and introducing a notion of Castelnuovo-Mumford regularity to establish splitting criteria.

Contribution

It extends cohomological splitting criteria and introduces Castelnuovo-Mumford regularity for vector bundles on complex product varieties.

Findings

Cohomological splitting conditions for rank 2 bundles

A cohomological characterization for vector bundles

Splitting criteria for vector bundles of arbitrary rank

Abstract

Here we consider the product of varieties with $n$-blocks collections . We give some cohomological splitting conditions for rank 2 bundles. A cohomological characterization for vector bundles is also provided. The tools are Beilinson's type spectral sequences generalized by Costa and Mir\'o-Roig. Moreover we introduce a notion of Castelnuovo-Mumford regularity on a product of finitely many projective spaces and smooth quadric hypersurfaces in order to prove two splitting criteria for vector bundle with arbitrary rank.