Holomorphic Sectional Curvature of Complex Finsler Manifolds

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.

Schwarz lemma from a Kähler manifold into a complex Finsler manifold

“…In [12], Shen and Shen obtained a Schwarz lemma from a compact complex Finsler manifold with holomorphic sectional curvature bounded from below by a negative constant into another complex Finsler manifold with holomorphic sectional curvature bounded above by a negative constant. In the case that the domain manifold is non-compact, Wan [14] obtained a Schwarz lemma from a complete Riemann surface with curvature bounded from below by a constant into a complex Finsler manifold with holomorphic sectional curvature bounded from above by a negative constant. The general case when the domain manifold is a complete non-compact complex manifold endowed with a strongly pseudoconvex complex Finsler metric is still open.…”

“…The general case when the domain manifold is a complete non-compact complex manifold endowed with a strongly pseudoconvex complex Finsler metric is still open. It seems that the method used in [14] does not work when the domain manifold has complex dimension ≥ 2.…”

A Schwarz lemma for weakly Kähler-Finsler manifolds

“…In [25], Shen and Shen obtained a Schwarz lemma from a compact complex Finsler manifold with holomorphic sectional curvature bounded from below by a negative constant into another complex Finsler manifold with holomorphic sectional curvature bounded above by a negative constant. In the case that the domain manifold is non-compact, Wan [28] obtained a Schwarz lemma from a complete Riemann surface with curvature bounded from below by a constant into a complex Finsler manifold with holomorphic sectional curvature bounded from above by a negative constant. The general case, i.e., when the domain manifold is a complete non-compact complex manifold endowed with a strongly pseudoconvex complex Finsler manifold, however, is still open.…”

“…The general case, i.e., when the domain manifold is a complete non-compact complex manifold endowed with a strongly pseudoconvex complex Finsler manifold, however, is still open. It seems that the method used in [28] does not work when the domain manifold has complex dimension ≥ 2.…”

Positivity preserving along a flow over projective bundle

“…In this section, we shall fix the notation and recall some basic definitions and facts on complex Finsler vector bundles, Griffiths positive (semipositive), and maximal principle for real (1, 1)-forms. For more details we refer to [2,4,9,10,12,13,15,19,21,23,33].…”