Chern forms of holomorphic Finsler vector bundles and some applications

Abstract

In this paper, we present two kinds of total Chern forms [Formula: see text] and [Formula: see text] as well as a total Segre form [Formula: see text] of a holomorphic Finsler vector bundle [Formula: see text] expressed by the Finsler metric [Formula: see text], which answers a question of Faran [The equivalence problem for complex Finsler Hamiltonians, in Finsler Geometry, Contemporary Mathematics, Vol. 196 (American Mathematical Society, Providence, RI, 1996), pp. 133–144] to some extent. As some applications, we show that the signed Segre forms [Formula: see text] are positive [Formula: see text]-forms on [Formula: see text] when [Formula: see text] is of positive Kobayashi curvature; we prove, under an extra assumption, that a Finsler–Einstein vector bundle in the sense of Kobayashi is semi-stable; we introduce a new definition of a flat Finsler metric, which is weaker than Aikou’s one [Finsler geometry on complex vector bundles, in A Sampler of Riemann–Finsler Geometry, MSRI Publications, Vol. 50 (Cambridge University Press, 2004), pp. 83–105] and prove that a holomorphic vector bundle is Finsler flat in our sense if and only if it is Hermitian flat.

Geodesic-Einstein metrics and nonlinear stabilities

“…By the Kobayashi correspondence (cf. [16], [12], [13]), a Finsler metric G on E induces a natural admissible metric on O P (E) (1). In this case, we can prove that the induced metric on O P (E) (1) is geodesic-Einstein if and only if G is Finsler-Einstein.…”

“…For any Finsler metric G on E, let φ be the corresponding Hermitian metric on the line bundle O P (E) (1), which is also an admissible metric on O P (E) (1) (cf. [16], [12], [13]). With respect to a holomorphic trivialization of E → M , by the standard procedure, one gets a local homogeneous holomorphic coordinate system {[ζ]|ζ = (ζ 1 , · · · , ζ r ) = 0} on the fibres of P (E).…”

“…Now the lemma is follows directly from (3.32) and the definitions of the Finsler-Einstein metric (cf. [12], Definition 3.1) and the geodesic-Einstein metric. Proof.…”

Holomorphic Sectional Curvature of Complex Finsler Manifolds

“…In this section, we shall fix notation and recall some basic definitions and facts on complex Finsler manifolds. For more details we refer to [ 1 , 4 , 5 , 10 , 13 , 18 , 27 ].…”

“…If G is a strongly pseudoconvex complex Finsler metric on M , then there is a canonical h-v decomposition of the holomorphic tangent bundle of (see [ 10 , §5] or [ 13 , §1]). In terms of local coordinates, Moreover, the dual bundle also has a smooth h-v decomposition with With respect to the h–v decomposition ( 2.12 ), the (1, 1)-form has the following decomposition.…”