Vector Bundles on Rational Homogeneous Spaces

2020/07/14 by Du, Rong, Fang, Xinyi, Gao, Yun
#Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2007.06816

Abstract

We consider a uniform r-bundle E on a complex rational homogeneous space X %over complex number field ℂ and show that if E is poly-uniform with respect to all the special families of lines and the rank r is less than or equal to some number that depends only on X, then E is either a direct sum of line bundles or δ_i-unstable for some δ_i. So we partially answer a problem posted by Muñoz-Occhetta-Solá Conde. In particular, if X is a generalized Grassmannian G and the rank r is less than or equal to some number that depends only on X, then E splits as a direct sum of line bundles. We improve the main theorem of Muñoz-Occhetta-Solá Conde when X is a generalized Grassmannian by considering the Chow rings. Moreover, by calculating the relative tangent bundles between two rational homogeneous spaces, we give explicit bounds for the generalized Grauert-Mülich-Barth theorem on rational homogeneous spaces.

Related

• Uniform bundles on generalised Grassmannians
• Uniform vector bundles over ℙ⁴
• Uniform (d+1)-bundle over the Grassmannian G(d,n)
• On uniform and nonhomogeneous vector bundles over Grassmannians
• Homogeneous ACM bundles on rational homogeneous spaces