Kähler-Einstein metrics with positive scalar curvature. (English) Zbl 0892.53027

Tian, Gang

Invent. Math. 130, No. 1, 1-37 (1997).

The author attacks the Calabi problem: give necessary and sufficient conditions for the existence of a Kähler-Einstein metric on a compact Kähler manifold \(M\) with \(c_1(M)>0 \). Generalizing the Futaki invariant, he defines the notion of weakly \(K\)-stability of a Kähler manifold \(M\). It is defined in terms of a degeneration of \(M\), that is an algebraic fibration \(\pi: W \to D\) over the unit disc \(D \subset \mathbb C\) without multiple fibers, such that \(M\) is biholomorphic to a fiber \(\pi^{-1}(z)\) for some \(z \in D\). The author proves that weakly \(K\)-stability of \(M\) is a necessary condition for the existence of a Kähler-Einstein metric on \(M\) and states a conjecture that this condition is also sufficient.
Using these results, the author disproves the long-standing folklore conjecture that the absence of holomorphic vector fields on a compact Kähler manifold \(M\) with \(c_1(M) >0\) is sufficient for the existence of a Kähler-Einstein metric. The counterexample is a smooth 3-dimensional submanifold of the Grassmanian \(G(4,7)\) first constructed by Iskovskih.
Under the assumption that the Kähler manifold \(M\) with Kähler form \(\omega\) has no nontrivial holomorphic vector field, the necessary and sufficient condition for existence of a Kähler-Einstein metric is given: some functional \(F_{\omega}\) on the space of smooth functions \(f\) with \(\omega + \partial \bar \partial f >0\) has to be proper.

Reviewer: D.V.Alekseevsky (Moskva)

MSC:

53C55	Global differential geometry of Hermitian and Kählerian manifolds
53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)

Keywords:

Kähler-Einstein metrics; positive scalar curvature; Fano manifolds; first Chern class; Futaki invariant