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# Communications in Analysis and Geometry

## Volume 12 (2004)

### Number 1

### The Futaki Invariant and the Mabuchi Energy of a Complete Intersection

Pages: 321 – 345

DOI: http://dx.doi.org/10.4310/CAG.2004.v12.n1.a15

#### Authors

#### Abstract

Let *M* be a compact complex Kähler manifold. If *c*_{1}*(M) = 0* or if *c*_{1}*(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 *c*_{1}*(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] ∈ H*^{2}*(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.