A classification of complex rank 3 vector bundles on \(\mathbb{C} P^5\)
type: Item
instance of: scholarly article
title: A classification of complex rank 3 vector bundles on \(\mathbb{C} P^5\)
author: Morgan Opie
published in: Advances in Mathematics
publication date: 2024-08-28T00:00:00Z
review text: The paper under review studies rank \(3\) topological vector bundles on \(\mathbb{C}P^{5}\). While Chern classes are enough to classify such bundles stably, it turns out that they do not suffice to provide unstable classification.\N\NSometimes Chern classes also provide unstable classification, namely for bundles on \(\mathbb{C} P^{n}\) of rank either \(\leq 1\) or \(\geq n\), but if the rank \(r\) satisfies \(1 < r <n\) there is no uniform answer.\N\NThe author summarizes earlier work on this topic. In particular, for rank \(2\) bundles on \(\mathbb{C}P^{3}\) there is a result of Atiyah and Rees which employs in addition to the Chern classes \(c_1\) and \(c_2\) a \(\mathbb{Z}/2\)-valued invariant \(\alpha\) obtained from a certain twisted characteristic class in \(KO\)-theory.\N\NThe main result of the paper is Theorem 1.4, a corollary of Theorems 1.7, 1.8 and 4.9, which are also of interest on their own.\N\NFirstly, Theorem 1.4 states that, given integers \(a_1\), \(a_2\) and \(a_3\) there exists a rank \(3\) bundle over \(\mathbb{C} P^{5}\) with Chern classes \(c_1\), \(c_2\) and \(c_3\) equal to \(a_1\), \(a_2\) and \(a_3\) respectively, if and only if Schwarzenberg identity \(S_5\) stated explicitly in Lemma 2.17 is fulfilled. Secondly, assuming \(S_5\), it says that depending on mod \(3\) properties of \(c_1\) and \(c_2\), there are \(3\) or \(1\) isomorphism classes of rank \(3\) bundles over \(\mathbb{C} P^{5}\) with fixed Chern classes \(c_1\), \(c_2\) and \(c_3\), and that these Chern classes together with a certain \(\mathbb{Z}/3\)-valued invariant \(\rho\) distinguish the isomorphism classes.\N\NTheorem 1.7, which can be viewed as a refinement of a part of Theorem 1.4 states that a certain geometrically defined action of \(\pi_{10} BU(3) \cong \mathbb{Z}/3\) on the set \(\mathcal{V}_{a_1,a_2,a_3}\) of isomorphism classes of rank \(3\) bundles on \(\mathbb{C} P^{5}\) with the Chern classes \(a_1\), \(a_2\) and \(a_3\) is transitive and describes the cases…
reviewed by: Tibor Macko
Mathematics Subject Classification ID: 55N20
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55P99
Mathematics Subject Classification ID: 55R15
Mathematics Subject Classification ID: 55R25
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 55S35
Mathematics Subject Classification ID: 55T25
Mathematics Subject Classification ID: 57R22
zbMATH DE Number: 7902802
zbMATH Keywords: vector bundles
zbMATH Keywords: unstable vector bundles
zbMATH Keywords: complex projective spaces
zbMATH Keywords: homotopy theory
MaRDI profile type: Publication
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DOI: 10.1016/J.AIM.2024.109878
description: scientific article; zbMATH DE number 7902802