Kodaira dimensions and hyperbolicity of nonpositively curved compact Kähler manifolds

F. Zheng
2002 Commentarii Mathematici Helvetici  
In this article, we prove that a compact Kähler manifold M n with real analytic metric and with nonpositive sectional curvature must have its Kodaira dimension, its Ricci rank and the codimension of its Euclidean de Rham factor all equal to each other. In particular, M n is of general type if and only if it is without flat de Rham factor. By using a result of Lu and Yau, we also prove that for a compact Kähler surface M 2 with nonpositive sectional curvature, if M 2 is of general type, then it
more » ... s Kobayashi hyperbolic. Mathematics Subject Classification (2000 ). Primary 53C55, Secondary 53C12. F. Zheng CMH tains rich examples. [Z1] and [Z2] provided some evidence on this point, especially in complex dimension 2. So we are not dealing with small or nearly empty set of specimen here. In the theory of nonpositively curved Riemannian manifolds, one of the main issues is to examine the fine distinction between the behavior of negatively curved and nonpositively curved manifolds. In the Kähler case, we propose the following conjectures which are simply based upon intuition: The Kodaira dimension of M is equal to the rank of its Ricci form, and is equal to the codimension of its Euclidean de Rham factor. In particular, M is of general type if and only if it has trivial Euclidean de Rham factor. 2. If M is of general type, then it is Kobayashi hyperbolic. 3. If M is of general type and is irreducible (i.e., no finite cover of M is biholomorphic to a product), and is not a locally Hermitian symmetric space of rank ≥ 2, then it satisfies the visibility axiom (i.e., the universal cover M does not contain any 2-flat). where M 0 is a locally Hermitian symmetric space, while for each 1 ≤ i ≤ k, M i is a irreducible (i.e., any finite cover of it is not biholomorphic to the product of Vol. 77 (2002) Kodaira dimensions 223
doi:10.1007/s00014-002-8337-z fatcat:utgv3qg33zddjbp5pzucajyavu