Curvatures of moduli space of curves and applications
Curvatures of moduli space of curves and applications
复制标题
曲线模空间的曲率及其应用
DOI：
--
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
S. Yau
中科院分区：
文献类型：
--
作者：
Kefeng Liu;Xiaofeng Sun;Xiaokui Yang;S. Yau
本文利用向量丛的直像层的曲率性质，研究了曲线模空间的几何性质。证明了亏格g>1的曲线的模空间（M_g，ω_{WP}）具有双Nakano负曲率和半Nakano负曲率，特别地，它具有非正的Riemann曲率算子和非正的复截面曲率.作为应用，我们证明了M_g$中的任意子流形，如果是A_g$中的全测地子流形且具有有限体积，则必是球商。我们还证明了在科代拉（形变）曲面上，虽然存在严格负全纯双截曲率的K“ahler度量，但不存在非正Riemann截曲率的K“ahler度量.
In this paper, we investigate the geometry of the moduli space of curves by using the curvature properties of direct image sheaves of vector bundles. We show that the moduli space $(M_g, omega_{WP})$ of curves with genus $g>1$ has dual-Nakano negative and semi-Nakano-negative curvature, and in particular, it has non-positive Riemannain curvature operator and also non-positive complex sectional curvature. As applications, we prove that any submanifold in $M_g$ which is totally geodesic in $A_g$ with finite volume must be a ball quotient. We also show that, on a Kodaira (deformation) surface, there is no K"ahler metric with non-positive Riemannain sectional curvature although it possesses K"ahler metrics with strictly negative holomorphic bisectional curvature.