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

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