An elementary proof of the irreducibility of the rank \(g\) Picard bundle
type: Item
instance of: scholarly article
title: An elementary proof of the irreducibility of the rank \(g\) Picard bundle
zbMATH Open document ID: 0899.14022
published in: Rocky Mountain Journal of Mathematics
publication date: 1998-11-08T00:00:00Z
full work available at URL: CONT26-4.html
review text: A (very) general principally polarized complex abelian variety \((A,\theta)\) satisfies Mattuck's property: its algebraic cohomology in degree \(2k\) is generated by \(\theta^k/k!\) [\textit{A. Mattuck}, Proc. Am. Math. Soc. 9, 88-98 (1958; Zbl 0118.37301)]. It follows from a result of Schur that the Chern polynomial of a vector bundle \(E\) on \(A\) with top Chern class \(\pm\theta^k/k!\) is irreducible over \({\mathbb{Q}}\); in particular, \(E\) is irreducible (i.e. is not extension of two vector bundles of smaller ranks). A general Jacobian is proved to satisfy Mattuck's property [this also follows from papers by \textit{S. Mori}, cf. Jap. J. Math., New Ser. 2, 109-130 (1976; Zbl 0339.14016) and 3, 105-109 (1977; Zbl 0379.14011)] hence the general rank \(g\) Picard bundle is irreducible.
Mathematics Subject Classification ID: 14K25
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 32J25
Mathematics Subject Classification ID: 14C22
zbMATH DE Number: 1036062
zbMATH Keywords: vector bundles
zbMATH Keywords: principally polarized complex abelian variety
zbMATH Keywords: algebraic curves
zbMATH Keywords: algebraic cohomology
zbMATH Keywords: general Jacobian
zbMATH Keywords: Picard bundle
author: David S. Yuen
MaRDI profile type: Publication
OpenAlex ID: W2019624877
cites work: Finite dimensional Teichmüller spaces and generalizations
cites work: Theta functions on Riemann surfaces
cites work: Q4108448
cites work: Inversion of Abelian integrals
cites work: Rank g Picard Bundles are Stable
cites work: Cycles on Abelian Varieties
cites work: Symmetric Products and Jacobians
cites work: Q4186470
DOI: 10.1216/RMJM/1181072006
description: scientific article; zbMATH DE number 1036062