Rank 2 vector bundles on higher dimensional projective manifolds
type: Item
instance of: scholarly article
title: Rank 2 vector bundles on higher dimensional projective manifolds
zbMATH Open document ID: 0892.14018
published in: Pacific Journal of Mathematics
publication date: 1997-09-16T00:00:00Z
review text: A well-known theorem of D. Gieseker and J. Li asserts that the moduli space of stable rank 2 vector bundles is reducible if ``the second Chern number'' is large enough and in particular the number of its irreducible components is bounded when \(c_2\) goes to infinity. An old result of Ein shows that this is not true on \(\mathbb{P}^3\). Here we use Ein's result to show that the same is true for a large class of higher-dimensional varieties. Fix a smooth complex projective \(n\)-fold \(X\), \(n\geq 5\), with \(H^1(X,{\mathcal O}_X)=0\). Then there is a smooth projective \(n\)-fold \(Y\) birational to \(X\) and \(H\in\text{Pic}(Y)\), \(H\) ample, with the following property: For any \(d\in\mathbb{Z}\) let \(M(Y,2{\mathcal O},d,H)\) be the moduli scheme of \(H\)-stable rank 2 vector bundles, \(E\), on \(Y\) with \(\text{det}(E)\cong{\mathcal O}_Y\) and \(c_2(E)\cdot H^{n- 2}=d\). Let \(m(Y,2,{\mathcal O},d,H)\) be the number of its irreducible components. Then \(\limsup_{d\to\infty} m(Y,2,{\mathcal O},d,H)=+\infty\).
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 1060825
zbMATH Keywords: number of irreducible components of the moduli space of stable rank 2 vector bundles
zbMATH Keywords: rank 2 vector bundles on higher dimensional projective manifolds
author: E. Ballico
MaRDI profile type: Publication
full work available at URL: pjm.1998.183.21
OpenAlex ID: W1984836707
DOI: 10.2140/PJM.1998.183.21
description: scientific article; zbMATH DE number 1060825