Indecomposable bundles on Cartesian products of odd projective spaces

Abstract

In this paper we construct indecomposable vector bundles associated to monads on Cartesian products of odd dimension projective spaces. Specifically we establish the existence of monads on $(\mathbb{P}^1)^{l_1}\times\cdots\times(\mathbb{P}^{2n+1})^{l_m}$. We prove stability of the kernel bundle and prove that the cohomology bundle is simple. We also prove the same for monads on $(\mathbb{P}^n)^2\times(\mathbb{P}^m)^2\times(\mathbb{P}^l)^2$ for an ample line bundle $\mathscr{L}=\mathcal{O}_X(\alpha,\alpha,\beta,\beta,\gamma,\gamma)$.