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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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