Ampleness equivalence and dominance for vector bundles

2017/06/22 by F. Laytimi, Laytimi, F., Werner Nahm +1
Mathematics · #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology #Geometric and Algebraic Topology

paper · doi:10.48550/arxiv.1706.07353

Abstract

Hartshorne in "Ample vector bundles" proved that E is ample if and only if \OOO_P(E)(1) is ample. Here we generalize this result to flag manifolds associated to a vector bundle E on a complex manifold X: For a partition a we show that the line bundle \it Q_a^s on the corresponding flag manifold Fl_s(E) is ample if and only if \SSS_aE is ample. In particular det Q on \itG_r(E) is ample if and only if \wedge ^rE is ample. We give also a proof of the Ampleness Dominance theorem that does not depend on the saturation property of the Littlewood-Richardson semigroup.

Related

• On k-ampleness equivalence
• Criteria for the ampleness of certain vector bundles
• Positivity of vector bundles on homogeneous varieties
• Ample vector bundles and branched coverings
• Partially ample line bundles and base loci