Abstract
In this paper, we get an inequality in terms of holomorphic sectional curvature of complex Finsler metrics. As applications, we prove a Schwarz Lemma from a complete Riemannian manifold to a complex Finsler manifold. We also show that a strongly pseudoconvex complex Finsler manifold with semi-positive but not identically zero holomorphic sectional curvature has negative Kodaira dimension under an extra condition.
Highlights
1 Introduction In this paper, we study the holomorphic sectional curvature of complex Finsler manifolds
This metric defines a Kobayashi pseudo-distance, and a complex manifold is called Kobayashi hyperbolic if the Kobayashi pseudo-distance is a distance in com
It is known that a complex Finsler manifold is Kobayashi hyperbolic if its holomorphic sectional curvature is bounded from above by a negative constant
Summary
We study the holomorphic sectional curvature of complex Finsler manifolds (see Definition 2.3). X. Wan compact strongly pseudoconvex Finsler manifold (M, G) with semi-positive but not identically zero holomorphic sectional curvature one has that κ(M) = −∞. Theorem 1.3 Let (M, G) be a compact strongly pseudoconvex complex Finsler manifold with semi-positive but not identically zero holomorphic sectional curvature, and satisfy ∂ ̄ G (P ) = 0, κ(M) = −∞. If G is a strongly pseudoconvex complex Finsler metric on M, there is a canonical h-v decomposition of the holomorphic tangent bundle T (T M)o of (T M)o (see [10, §5] or [13, §1]). For a strongly pseudoconvex complex Finsler metric G, one can define the holomorphic sectional curvature by KG vi vj vk vl = 2 Ri jkl G2.
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