Vector Bundles On Rational Homogeneous Spaces

Vector bundles are important research objects in algebraic geometry.In this paper,we mainly study algebraic vector bundles over an algebraically closed field.Grothendieck's theo-rem shows that every vector bundle on the projective line splits as a direct sum of line bundles.It is natural to consider restrictions of vector bundles to lines in order to study them on projec-tive varieties covered by lines.In this paper,we consider two kind of vector bundles on rational homogeneous spaces:uniform vector bundles and semistable vector bundles.The following results are obtained.1.We classify uniform r(r?min{d,n-d})-bundles on Grassmannians G(d,n)in positive characteristic.2.We give a classification of uniform low-rank vector bundles on generalized Grassmannians in characteristic zero.3.We characterize the general splitting types of semistable vector bundles on rational ho- mogeneous spaces in characteristic zero and give the generalized Barth-Grauert-Mülich theorem.4.We obtain some constraints of normal bundles with the exceptional sets of simple small resolutions as generalized Grassmannians.In particular,we partially answer the classification problem of uniform low-rank vector bundles on rational homogeneous spaces posted by Mu?oz-Occhetta-SoláConde.