Manchester Geometry Seminar 2008/2009

A method of Kobayashi constructs an Einstein metric with positive Ricci curvature on a circle bundle over a complex Fano manifold that admits a Kähler-Einstein metric. Boyer and Galicki generalized the method of Kobayashi to a Seifert bundle over a Fano orbifold that admits an orbifold Kähler-Einstein metric. But the existence of Kähler-Einstein metrics on Fano orbifolds is a subtle problem that is still unsolved (even for Fano manifolds). For smooth two-dimensional Fano manifolds (del Pezzo surfaces), this problem has been completely solved by Gang Tian in 1990.

For Fano orbifolds, we know many obstructions to the existence of a Kähler-Einstein metric, which are due to Matsushima, Futaki, Tian, Yau, Sparks, Donaldson and Thomas. For a Fano orbifold V, it has been conjectured by Yau that V admits a Kähler-Einstein metric if and only if V is stable in a certain sense. Proving this conjecture is currently a major research programme in geometry (the talk of Simon Donaldson on Miles60 conference in London (July, 2008) is exactly about this conjecture). See http://arxiv.org/abs/0801.4179 and http://arxiv.org/abs/0803.0985. But even for smooth three-dimensional Fano manifolds, we know little about the existence of a Kähler-Einstein metric. The only known sufficient condition for the existence of a Kähler-Einstein metric on a Fano orbifold is due to Tian, Siu, Nadel, Kollar and Demailly. This condition can be formulated in terms of the so-called alpha-invariant of Tian (or so-called global log canonical threshold for Fano orbifolds). This holomorphic invariant plays important role in Kähler geometry (existence of a Kähler-Einstein metric, convergence of the Kähler-Ricci flow, etc). Surprisingly, this invariant also plays important role in birational geometry and singularity theory. We describe this invariant and its application in geometry.