Notes on homogeneous vector bundles over complex flag manifolds

Let P be a parabolic subgroup of a semisimple complex Lie group G defined by a subset Σ of simple roots of G, and let Eφ be a homogeneous vector bundle over the flag manifold G/P corresponding to a linear representation φ of P . Using Bott’s theorem, we obtain sufficient conditions on φ in terms of the combinatorial structure of Σ for some cohomology groups of the sheaf of holomorphic sections of Eφ to be zero. In particular, we define two numbers d(P ), l(P ) ∈ N such that for any φ obtained by natural operations from a representation of dimension less than d(P ) the q-th cohomology group of Eφ is zero for 0 < q < l(P ). We prove also that in this case that the vector bundle Eφ is rigid. Let E be a holomorphic vector bundle over a connected compact complex manifold M . Then there is a natural homomorphism of complex Lie groups μ : AutE → BihM , where AutE is the automorphism group of the vector bundle and BihM is the group of all biholomorphic transformations of M . The bundle E is said to be homogeneous if the action μ of AutE on M is transitive. Assume that we have a homomorphism Φ: G → AutE such that the action μΦ of G on M is transitive. Then E is homogeneous; we say that E is homogeneous with respect to G. Let o ∈ M and let P = Go be the stabilizer of o in G. Then P acts in a natural way on the fibre E = Eo, that is, we have a holomorphic linear representation φ : P → GL(E). It is known that the bundle E is uniquely determined by the group G, the subgroup P , and the representation φ, which can all be arbitrary. We denote by Eφ the homogeneous vector bundle over M = G/P defined by a representation φ of P . Note also that corresponding to standard tensor operations on representations there are similar operations on vector bundles with a fixed base. In particular, (1) Eφ∗ = E ∗ φ, Eφ1+φ2 = Eφ1 ⊕ Eφ2 , Eφ1φ2 = Eφ1 ⊗ Eφ2 . For a Lie group G denote by G the identity component of G. We consider the case when M is a flag manifold, that is, when M is homogeneous and the stabilizers (BihM)x ⊂ (BihM) , x ∈ M , are parabolic subgroups (see [1, 6]). Then the group BihM and all its transitive subgroups are semisimple Lie groups. 1991 Mathematics Subject Classification. Primary 32M10, 32L10, 17B10, 17B20.