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ON HYPERSURFACES OF A COMPLEX

On Kahler submanifolds of a complex projective space, J-I Hano [5] has studied complete intersections of hypersurfaces in a complex projective space and proved that if a complete intersection M of hypersurfaces is an Einstein manifold with respect to the induced metric then M is a complex projective space or a complex quadric. The purpose of this note is to investigate hypersurfaces of a complex Grassmann manifold by using Hano's method. Let Gm+n n(C) denote the complex Grassmann manifold of τz-ρlanes in C . Let X be a compact complex hypersurface of Gm+n n(C). Then X defines a positive divisor on Gm+n n(C) and hence a holomorphic line bundle {X} on Gm+nn(C). We denote by c(X) the Chern class of the line bundle {X}. Since the second cohomology group H\Gm+n n(C)> Z) is isomorphic to Z, we can write c(X)= a(X)-σ, where a(X)^N and σ is a generator of H\Gm+nn(C)y Z). We call a(X) the degree of X, We equip an hermitian inner product on C. The complex Grassmann manifold Gm+nn(C) has a Kahler metric invariant under the action of the unitary group U(m-\-ri). Moreover we may assume that m^tn. Under these notations, we have a following Theorem.