Some properties of vector bundles on the flag variety Fl(r,s;n)
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''.
Chern classes of tautological vector bundles
scientific article; zbMATH DE number 3851258