Vector Bundle construction via Monads on multiprojective Spaces

TL;DR

This paper demonstrates the existence and stability of monads on multiprojective spaces, generalizing instanton bundles and constructing specific morphisms, advancing the understanding of vector bundles in algebraic geometry.

Contribution

It establishes the existence of monads on multiprojective spaces and proves stability and simplicity of associated vector bundles, extending previous results to broader classes of spaces.

Findings

Existence of monads on multiprojective spaces.

Stability of the kernel bundle.

Simplicity of the cohomology vector bundle.

Abstract

In this paper we establish the existence of monads on multiprojective spaces $X=\mathbb{P}^{2n+1}\times\mathbb{P}^{2n+1}\times\cdots\times\mathbb{P}^{2n+1}$. We prove stability of the kernel bundle which is a dual of a generalized Schwarzenberger bundle associated to the monads and prove that the cohomology vector bundle is simple, a generalization of instanton bundles. Next we construct monads on $\mathbb{P}^{a_1}\times\cdots\times\mathbb{P}^{a_n}$ and prove stability of the kernel bundle and that the cohomology vector bundle is simple. Lastly, we construct the morphisms that establish the existence of monads on $\mathbb{P}^1\times\cdots\times\mathbb{P}^1$.