## Osaka Journal of Mathematics

### 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 $\mathbb{P}^{n}$.

#### Article information

Source
Osaka J. Math., Volume 48, Number 3 (2011), 589-612.

Dates
First available in Project Euclid: 26 September 2011

https://projecteuclid.org/euclid.ojm/1317044937

Mathematical Reviews number (MathSciNet)
MR2837671

Zentralblatt MATH identifier
1228.53087

#### Citation

Wong, Pit-Mann; Zhong, Chunping. On weighted complex Randers metrics. Osaka J. Math. 48 (2011), no. 3, 589--612. https://projecteuclid.org/euclid.ojm/1317044937

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