Some properties of vector bundles on the flag variety Fl(r,s;n)
type: Item
instance of: scholarly article
title: Some properties of vector bundles on the flag variety Fl(r,s;n)
zbMATH Open document ID: 0536.14013
author: Tohsuke Urabe
published in: Tokyo Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: The first attempt to understand vector bundles on the flag varieties by the author is this article. He treats the flag variety F parametrizing two linear spaces in inclusion relation. The first one third of the article is devoted to constructing the component \(\hat X\) of the Hilbert scheme of F corresponding to ''straight lines'' in F. For a given vector bundle E on F, by restricting E to any ''straight line'' contained in F and by studying properties of the restricted bundles, we can analyse global properties of E. The construction of \(\hat X\) is indispensable for this purpose. It turns out that \(\hat X\) carries also natural tautological vector bundles. In the middle part the author shows that Chern classes of tautological vector bundles on F and those on \(\hat X\) have very systematic correspondence. This is an interesting fact. In the last section he shows that the set J of jumping lines coincides with the image of the second Chern class of E by the Gysin homomorphism in the Chow group under a certain hypothesis. Here J is the set of points in \(\hat X\) such that the restriction of E to the corresponding ''line'' is different from that of the general ''line''.
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 3851258
zbMATH Keywords: vector bundles on flag variety
zbMATH Keywords: Hilbert scheme
zbMATH Keywords: Chern classes of tautological vector bundles
zbMATH Keywords: jumping lines
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270213875
OpenAlex ID: W1994034421
DOI: 10.3836/TJM/1270213875
description: scientific article; zbMATH DE number 3851258