Monads on Cartesian products of projective spaces

TL;DR

This paper constructs and analyzes monads on specific Cartesian products of projective spaces, demonstrating their stability, simplicity, and existence under certain conditions, extending the theory of monads in algebraic geometry.

Contribution

It introduces new monads on complex product spaces, proving their stability, simplicity, and establishing their existence in broader configurations.

Findings

Existence of monads on special Cartesian products of projective spaces.

Proved stability of the kernel bundle associated with these monads.

Established simplicity of the cohomology vector bundle.

Abstract

In this paper we establish the existence of monads on special Cartesian products of projective spaces. Special in the sense that we mimick monads on instanton bundles. We construct monads on $\mathbb{P}^1\times\cdots\times\mathbb{P}^1\times\mathbb{P}^3\times\cdots\times\mathbb{P}^3\times\mathbb{P}^5\times\cdots\times\mathbb{P}^5$. We proceed to prove stability of the kernel bundle associated to the monad and simplicity of the cohomology vector bundle. Lastly we establish the existence of monads on $\mathbb{P}^{a_1}\times\cdots\times\mathbb{P}^{a_n}$ where $a_1<a_2<\ldots<a_n$, alternating even and odd or at least $a_i$ $0<i\leq{n}$ is odd.