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 $\mathscr{L}=\mathcal{O}_X(\alpha,\alpha,\beta,\beta,\gamma,\gamma)$.