On the nullities of Kahler C-spaces in $P_{N}(C)$

Abstract

Let Mbe a Kahler C-space which Is holomorphically and isometricallyimbedded in an N-dimensional complex projectivespace Pjv(C). Then M is a minimal submanifold of Pn(C). Let na(M) be theanalytic nullityof M which was definedin [2]. We know that the nullityn{M) of M is equal tona{M) if M is a Hermitian symmetric space (Kimura [2]). In this note we prove that w(M)-=na(M) forany Kahler C-space M. By a theorem of Simons [5], the nullity of a Kahler submanifold coincides with the real dimension of the space of holomorphic sectionsof a normal bundle of the submanifold. Put M=G/U where G is a complex semi-simple Lie group and U is a parabolic subgroup of G. By a resultof Nakagawa and Takagi [4],we know that every imbedding of M in Pn(C) is induced by a holomorphic linear representationof G. From thisresultwe see that the normal bundle N(M) over M is a homogeneous vector bundle. We prove Theorem 1 which generalizes the generalized Borel-Weil theorem of Bott [1]. Applying the theorem to calculatethe dimension of the space of holomorphic sectionsof N(M) and prove that n(M) =na(M). The auther proved the above result before Proffesor Takeuchi gave another proofof it. His proof does not use Theorem 1 and is more simple than our proof (c.f.Takeuchi [6]). §1. The generalization of Bott'sresultLet G be a simply connected compact semi-simple Lie group with Lie algebra 8. Take a Cartan subalgebra I)of 0. Denoto by A therootsystem ofgwith respect to fy We fixa linear order on the realvector space spaned by theelements ereA. Let A+ (resp.A~)be the setof allpositive(resp.negative)roots. LetIT={au･･･,ai) be the fundamental root system, where /is therank of g and IL.be a subsystem ofII. We put