Simple Cohomology bundles on multiprojective spaces

TL;DR

This work analyzes vector bundles arising from monads on the multiprojective space $X = (\mathbf P^{n_1})^2\times\cdots\times(\mathbf P^{n_s})^2$ with ample polarization $\mathscr{L}=\mathcal{O}_X(\alpha_1,\alpha_1,\ldots,\alpha_s,\alpha_s)$. It constructs explicit linear monads built from blocks $\mathscr{G}_{\alpha_i}$ and proves that the kernel bundle $K=\ker(g)$ is $\mathscr{L}$-stable, while the cohomology bundle $E=\ker g/\mathrm{im} f$ is simple, giving a precise rank formula $\mathrm{rk}(E)=2\bigl(n_1+\cdots+n_s+k(s-1)\bigr)$. The approach relies on a generalized Hoppe criterion for polycyclic varieties and vanishing results for exterior powers, enabling stability of $K$ and simplicity of $E$. This extends prior results on monads from projective spaces to products of squared projective spaces and lays groundwork for moduli questions of such bundles on multiprojective spaces. The results have potential implications for constructing stable and simple vector bundles in more complex ambient varieties and contribute to the broader understanding of monad techniques in higher-dimensional settings.

Abstract

We prove stability of the kernel bundle and prove that the cohomology bundle is simple for vector bundles associated to monads on $X = (\mathbb{P}^{n_1})^2\times\cdots\times(\mathbb{P}^{n_s})^2$ for an ample line bundle $\mathscr{L}=\mathcal{O}_X(α_1,α_1,\cdots,α_s,α_s)$.

Theorems & Definitions (21)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4: Generalized Hoppe Criterion
• Lemma 2.5
• Theorem 2.7: 13, Theorem 4.1, page 131
• Lemma 2.8: 9, Lemma 2.11
• Lemma 2.9
• Lemma 2.10: 11, Theorem 2.4