复Finsler向量丛的曲率

在本项目中，本人主要围绕复Finsler向量丛的曲率性质进行研究。对于一个复Finsler向量丛，可以定义Kobayashi曲率，它是tautological 线丛第一陈形式的水平部分，我们将考虑Finsler向量丛的对偶性问题，即全纯向量丛E上存在一个正Kobayashi曲率的复Finsler度量是否等价于在其对偶丛上存在一个具有负Kobayashi曲率的复Finsler度量；对于全纯切丛，Mori证明了具有ample切丛的紧复流形必双全纯于复射影空间，我们将使用复Finsler几何的方法理解Mori的定理；最后，Wu-Yau 猜测对于一个Kahler流形如果射影丛上的tautological线丛的第一陈类在tautological 方向是负的，那么它的典范线丛是ample的， 我们围绕这一猜测展开对Finsler流形全纯截曲率的研究。

In this project, we will study the curvature of complex Finsler vector bundles. For a complex Finsler vector bundle, one can define Kobayashi curvature, it is the horizontal part of first Chern form of tautological line bundle, we will consider the dual problem on complex Finsler vector bundles, i.e. a holomorphic vector bundle E admits a Finsler metric with positive Kobayashi curvature if and only if its dual vector bundle admits a Finsler metric with negative Kobayashi curvature; For the holomorphic tangent bundle, Mori proved that the manifold must be biholomorphic to a complex projective space if its tangent bundle is ample, we will study Mori's theorem by using the method of complex Finsler geometry. Lastly, Wu-Yau conjectured that for a compact Kahler manifold such that over the projectivized tangent bundle, the first Chern class of the tautological line bundle is negative in the tautological direction, then the canonical line bundle is ample, we will study the holomorphic sectional curvature of complex Finsler manifold around this conjecture.