A Cohomological criterion for the splitting of vector bundles on $\mathbb{P}^{n_1}\times\cdots\times\mathbb{P}^{n_s}$

TL;DR

This paper develops cohomological criteria to determine when vector bundles on multiprojective spaces split into simpler components, extending existing vanishing cohomology conditions to more complex product spaces.

Contribution

It introduces a new cohomological criterion for splitting vector bundles on multiprojective spaces and generalizes vanishing cohomology conditions to these spaces.

Findings

Established a cohomological criterion for splitting vector bundles on multiprojective spaces.

Generalized vanishing cohomology criteria for vector bundles on product projective spaces.

Provided theoretical tools for analyzing vector bundle decompositions in complex geometric settings.

Abstract

In this paper we study the cohomological criterion for the splitting of vector bundles on multiprojective spaces $\mathbb{P}^{n_1}\times\ldots\times\mathbb{P}^{n_s}$. We also give a generalization of vanishing cohomological criteria for vector bundles on $\mathbb{P}^{n}\times\ldots\times\mathbb{P}^{n}$.