Regularity and cohomological splitting conditions for vector bundles on multiprojective spaces
type: Item
instance of: scholarly article
title: Regularity and cohomological splitting conditions for vector bundles on multiprojective spaces
zbMATH Open document ID: 1246.14056
author: E. Ballico
author: Francesco Malaspina
published in: Journal of Algebra
publication date: 2012-04-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/0802.0960
review text: The authors consider a general product of two projective spaces \(X=\mathbb P^n\times \mathbb P^m\) and study the cohomology of vector bundles over \(X\). They start with a new definition of regularity, for coherent sheaves on \(X\), which modifies a previous definition given by \textit{J. W. Hoffman} and \textit{H. H. Wang} [Adv. Geom. 4, No. 4, 513--536 (2004; Zbl 1063.13001)]. In details, \(F\) is \((p,p')\)-regular if \(H^i(F(p,p')\otimes \mathcal O (j,k))=0\) for \(-n\leq j\leq 0\), \(-m\leq k\leq 0\) and \(j+k=-i\). The authors prove several properties of regular sheaves on the product \(X\), most of which are similar to the properties of Castelnuovo-Mumford regular sheaves on projective spaces . In the spirit of the new definition, the authors extend Horrocks splitting criterion for vector bundles. Indeed, they show that a rank \(r\) bundle \(E\) on \(X\) splits in a sum of line bundles \(E=\oplus^r\mathcal O(t_s,t_s)\) if and only if for \(i=1,\dots,m+n-1\) and for all \(t\), the cohomology groups \(H^i(E(t,t)\otimes \mathcal O (j,k))\) vanish whenever \(-n\leq j\leq 0\), \(-m\leq k\leq 0\) and \(j+k=-i\).
reviewed by: Luca Chiantini
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14F17
zbMATH DE Number: 6022752
zbMATH Keywords: vector bundles
zbMATH Keywords: multiprojective spaces
MaRDI profile type: Publication
OpenAlex ID: W2115631681
arXiv ID: 0802.0960
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DOI: 10.1016/J.JALGEBRA.2011.08.015
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description: scientific article; zbMATH DE number 6022752