Algebras of curvature forms on homogeneous manifolds

Abstract

Let C(X) be the algebra generated by the curvature two-forms of standard holomorphic hermitian line bundles over the complex homogeneous manifold X = G=B. The cohomology ring of X is a quotient of C(X). We calculate the Hilbert polynomial of this algebra. In particular, we show that the dimension of C(X) is equal to the number of independent subsets of roots in the corresponding root system. We also construct a more general algebra as- sociated with a point on a Grassmannian. We calculate its Hilbert polynomial and present the algebra in terms of generators and relations. 1. Homogeneous Manifolds In this section we remind the reader the basic notions and notation related to homogeneous manifolds G=B and root systems, as well as x our terminology. Let G be a connected complex semisimple Lie group and B its Borel subgroup. The quotient space X = G=B is then a compact homogeneous complex manifold. We choose a maximal compact subgroup of G and denote by T = K B its maximal torus. The group acts transitively on X.T husX can be identied with the quotient space K=T. By g we denote the Lie algebra of G and by h g its Cartan subalgebra. Also denote bygR g the real form ofg such that igR is the Lie algebra of K. Analogously,hR =h\gR and ihR is the Lie algebra of the maximal torus T.T he root system associated with g is the set of nonzero vectors (roots) 2h for which the root spaces