Monads on multiprojective spaces and associated vector bundles

TL;DR

This paper proves the existence of monads on products of projective spaces and constructs associated vector bundles, analyzing their stability and simplicity, which advances understanding of vector bundle theory on multiprojective spaces.

Contribution

It establishes the existence of monads on multiprojective spaces and demonstrates the stability and simplicity of the resulting vector bundles.

Findings

Monads exist on Cartesian products of projective spaces.

Constructed vector bundles are proven to be stable.

Constructed vector bundles are proven to be simple.

Abstract

In this paper we establish the existence of monads on Cartesian products of projective spaces. We construct vector bundles associated to monads on $\mathbb{P}^{a_1}\times\mathbb{P}^{a_1}\times\mathbb{P}^{a_2}\times\mathbb{P}^{a_2}\times\cdots\times\mathbb{P}^{a_n}\times\mathbb{P}^{a_n}$. Once the monad on $X$ exists the next natural question is if the cohomology vector bundle associated to these monads are simple or not. We study these vector bundles associated to monads on $X$ and prove their stability and simplicity.