On weighted complex Randers metrics

Abstract

In this paper we introduce the weighted complex Randers metric $F=h+\sum_{i=1}^{m}\lvert B_{i}\rvert^{1/i}$ on a complex manifold $M$, here $h$ is a Hermitian metric on $M$ and $B_{i}$, $i=1,\ldots, m$ are holomorphic symmetric forms of weights $i$ on $M$, respectively. These metrics are special case of jet metric studied in Chandler--Wong [6]. Our main theorem is that the holomorphic sectional curvature $\mathrm{hbsc}_{F}$ of $F$ is always less or equal to $\mathrm{hbsc}_{h}$. Using this result we obtain a rigidity result, that is, a compact complex manifold $M$ of complex dimension $n$ with a weighted complex Randers metric $F$ of positive constant holomorphic sectional curvature is isomorphic to $\mathbb{P}^{n}$.