Indecomposable bundles on Cartesian products of odd projective spaces

TL;DR

The paper constructs indecomposable low-rank vector bundles on multiprojective varieties by realizing them as cohomologies of explicitly built monads. It centers on monads on $X=(\mathbf P^1)^{l_1}\times(\mathbf P^3)^{l_2}\times\cdots\times(\mathbf P^{2n+1})^{l_m}$, proving the kernel is stable and the cohomology bundle is indecomposable, and then extends the analysis to $X=(\mathbf P^{n})^2\times(\mathbf P^{m})^2\times(\mathbf P^{l})^2$ with a given polarization, obtaining similar indecomposability results. The construction employs a Segre embedding to produce explicit linear monads with morphisms built from Segre coordinates, along with stability checks via a generalized Hoppe criterion on polycyclic varieties. These results advance the catalog of indecomposable bundles on multiprojective spaces and generalize prior monad constructions by Maingi and others, leveraging precise cohomological vanishing arguments and stability analyses. Overall, the work provides explicit, verifiable monad data yielding simple, indecomposable cohomology bundles on new families of multiprojective varieties.

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 $\mathscr{L}=\mathcal{O}_X(α,α,β,β,γ,γ)$.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5: Generalized Hoppe Criterion
• Lemma 2.7
• Theorem 2.9: Künneth formula
• Theorem 2.10: 15, Theorem 4.1, page 131