Communications in Analysis and Geometry
Volume 12 (2004)
The Futaki Invariant and the Mabuchi Energy of a Complete Intersection
Pages: 321 – 345
Let M be a compact complex Kähler manifold. If c1(M) = 0 or if c1(M) < 0, then it is known by the work of Yau [Y78] and Yau, Aubin [Y78], [A78] that M has a Kähler-Einstein metric. If c1(M) > 0, then there are obstructions to the existence of such a metric, and here the guiding conjecture is that formulated by Yau in [Y93], which says that M has a Kähler-Einstein metric if and only if M is stable in the sense of geometric invariant theory.
An important obstruction to the existence of Kähler-Einstein metric is the invariant of Futaki [F83], which is a map F : η(M) → C with the following properties: F is a Lie algebra character on the space η(M) of holomorphic vector fields, which depends only on the cohomology class [ω] ∈ H2(M). The vanishing of F is a necessary condition for the existence of a Kähler-Einstein metric on M. However, it is not a sufficient condition: in [T97], Tian gives an example of a manifold with η(M) = 0 (so that the Futaki invariant vanishes trivially) with the property that M has no Kähler- Einstein metric.