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Complex Finsler metrics

In this paper we describe an approach to complex Finsler metrics suitable to deal with global questions, and stressing the similarities between hermitian and complex Finsler metrics. Let $F$ be a smooth complex Finsler metric on a complex manifold $M$, and assume that the indicatrices of $F$ are strongly pseudoconvex -- we shall say that $F$ itself is strongly pseudoconvex. The vertical bundle $\cal V$ is the kernel of the differential of the canonical projection of the holomorphic tangent bundle of $M$. Using $F$, it is possible to endow $\cal V$ with a hermitian metric; let $D$ be the Chern connection associated to this metric. It turns out that there is a canonical way to build starting from $D$ a horizontal bundle $\cal H$, as well as a bundle isomorphism $Θ\colon{\cal V}\to\cal H$. Using $Θ$ we may transfer both the metric and the connection on $\cal H$; furthermore, there is a canonical isometric embedding $χ$ of the holomorphic tangent bundle of $M$ into $\cal H$. Our idea is that the Finsler geometry of $M$ can be studied applying standard hermitian techniques to $\cal H$ using $χ$ to transfer back and forth problems and solutions. To support this claim, in this paper we di