A Cohomological criterion for the splitting of vector bundles on $\mathbb{P}^{n_1}\times\cdots\times\mathbb{P}^{n_s}$

TL;DR

This paper advances Horrocks-type splitting criteria for vector bundles to multiprojective spaces X = P^{n1} × ... × P^{ns}. It develops and employs regularity concepts on multiprojective spaces, together with the Kunneth formula and Koszul complexes, to derive vanishing conditions that precisely characterize when a vector bundle splits into a sum of line bundles twisted by diagonal multiples O_X(l,...,l). The main contributions are two theorems giving necessary and sufficient cohomology vanishing criteria for splitting into line-bundle sums with controlled multi-degree, plus an equivalent formulation in the equal-n setting. The results extend Miyazaki's biprojective criteria to the multiprojective context, enriching the understanding of cohomological criteria for decomposability in complex product spaces.

Abstract

In this paper we study the cohomological criterion for the splitting of vector bundles on multiprojective spaces $\mathbb{P}^{n_1}\times\ldots\times\mathbb{P}^{n_s}$. We also give a generalization of vanishing cohomological criteria for vector bundles on $\mathbb{P}^{n}\times\ldots\times\mathbb{P}^{n}$.

Theorems & Definitions (17)

• Theorem 1.2
• Theorem 1.3
• Lemma 1.4
• Definition 2.1
• Definition 2.2
• Theorem 2.3: Künneth formula
• Lemma 2.4
• Remark 2.5
• Definition 2.6
• Proposition 2.7