全纯纤维全空间上负曲率Finsler度量的研究

Finsler Metrics of Negative Curvature on the Total Space of a Holomorphic Fibration

摘要: 本文旨在紧全纯纤维化的全空间上，构造具有负全纯截面曲率的复Finsler度量。设$p:X\to B$ 为紧复流形间的全纯纤维化，假设底空间$B$ 具有负全纯截面曲率的Hermitian度量$h_B$ ，全空间$X$ 具有光滑强拟凸复Finsler度量$G$ ，且$G$ 在每个纤维$X_t=p^{-1}\left(t\right)$ 上的限制具有负全纯截面曲率。本文采用混合叠加构造$F_k=G+kA=G+k{p}^{\ast }{h}_B$ ，证明：当参数$k$ 充分大时，$F_k$ 是$X$ 上光滑强拟凸复Finsler度量，并具有负全纯截面曲率。

Abstract: This paper constructs complex Finsler metrics with negative holomorphic sectional curvature on the total space of a compact holomorphic fibration. Let $p:X\to B$ be a holomorphic fibration between compact complex manifolds. Assume that the base $B$ admits a Hermitian metric $h_B$ of negative holomorphic sectional curvature, and that $X$ is equipped with a smooth strongly pseudoconvex complex Finsler metric $G$ whose restriction to every fiber $X_t=p^{-1}\left(t\right)$ has negative holomorphic sectional curvature. We consider the mixed metric $F_k=G+kA=G+k{p}^{\ast }{h}_B$ . Prove that, for all sufficiently large $k$ , $F_k$ is a smooth strongly pseudoconvex complex Finsler metric on $X$ with negative holomorphic sectional curvature.