Rationally Connected Varieties

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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.