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K-stability of constant scalar curvature Kähler manifolds. (English) Zbl 1181.53060
Let $$(X,L)$$ be a polarised manifold. Recently, it has been shown that if one can find a constant scalar curvature Kähler metric $$g$$ on $$X$$ whose $$(1,1)$$-form $$\omega_{g}$$ belongs to the cohomology class $$c_{1}(L)$$ then $$(X,L)$$ is semi-stable, in a number of senses. The paper under review is concerned with Donaldson’s algebraic $$K$$-stability. It is shown that a polarized manifold with a constant scalar curvature Kähler metric and discrete automorphisms is $$K$$-stable.

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 14L24 Geometric invariant theory
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##### References:
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