Rationally Connected Varieties

本文的主要目的是研究代数几何与微分几何和数论的相互作用。首先，该提案旨在进一步探索法诺轨道上的kaehler -Einstein度量和奇维流形上的爱因斯坦度量之间的联系，奇维流形可以实现为代数变体上的轨道圆束。另一个目的是研究各种几何（例如群作用或Gromov—Witten不变量）和关于在特殊域上定义的点和有理曲线存在性的算术问题之间的相互作用。如果我们用可拉伸的材料做一个香蕉或一个盒子，我们开始把它吹起来，最终它开始看起来像一个球体。这是因为球体是在空间中实现这种形状的最对称的方式，其中表面张力分布最均匀。对于任意形状和更高维度，爱因斯坦写下了最佳实现的某些条件。这些现在被称为爱因斯坦度量。最近（与Boyer和Galicki一起），我发现了一种新的方法，可以为许多不同的形状构建这样的爱因斯坦度量。该提案的一个主要目的是了解该方法的适用性、范围和局限性。

The main aim of the proposal is to study interactions ofalgebraic geometry with differential geometry and with number theory.First, the proposal aims to explore further the connections betweenKaehler--Einstein metrics on Fano orbifolds and Einstein metricson odd dimensional manifolds that can be realized as orbifoldcicrle bundles over algebraic varieties. Another aim is to studythe interactions between the geometry of a variety (for instancegroup actions or Gromov--Witten invariants) and arithmetic problemsabout the existence of points and rational curves defined overspecial fields.If we make a banana or a box out of stretchable materialand we start blowing it up, eventually it starts to looklike a sphere. This happens because the sphere is themost symmetric way of realizing this shape in space, wherethe surface tension is most evenly distributed. For arbitrary shapes and in higher dimensions, Einstein wrote downcertain conditions for the optimal realizations. These are now calledEinstein metrics. Recently (with Boyer and Galicki) I have discovereda new method to contruct such Einstein metrics for many differentshapes. A main aim of the proposal is to understand the applicability, scope and limitations of this method.