Showing papers on "Complex dimension published in 1987"

Uniqueness theorems of Hermitian metrics of seminegative curvature on quotients of bounded symmetric domains

Abstract: In the theory of locally symmetric spaces of negative Ricci curvature it has been a classical problem to study the extent to which the topology of the manifold determines the geometry, and, in the Hermitian case, the complex structure. The rigidity theorem of Mostow [26] asserts that for a compact locally symmetric Riemannian manifold of negative Ricci curvature, the manifold is determined up to isometry and normalizing constants by its fundamental group among the class of such manifolds, with the obvious exceptions involving compact Riemann surfaces. The same theorem for locally symmetric Riemannian manifolds of finite volume and negative Ricci curvature was proved by Prasad [27] in case of rank 1 and included in the super-rigidity theorem of Margulis [13] in the case of rank ? 2. In the class of compact locally symmetric Hermitian manifolds of negative Ricci curvature, Calabi-Vesentini [5] and Borel [3] proved vanishing theorems of certain cohomology groups which imply in particular that for X locally irreducible of complex dimension> 2, there exists no non-trivial deformation of X as a complex manifold. In this direction Siu ([32], [33]) proved the strong rigidity of the Kdhler manifold X in the sense that any compact Kkhler manifold M homotopic to X is necessarily biholomorphic or conjugate-biholomorphic to X. If M is also locally symmetric by assumption then it follows from the uniqueness of Kihler-Einstein metrics of negative Ricci curvature (Yau [37]) that the (conjugate-) biholomorphism is in fact an isometry up to a normalizing constant. It is thus natural to ask

On the codegree of negative multiples of the Hopf bundle

Abstract: Let H be the Hopf line bundle over the complex projective space of complex dimension k – 1. We determine the codegree of the virtual bundle –nH in the range l ≦ n ≦ k. This codegree has geometric significance as a stable James number.