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In differential geometry, a constant scalar curvature K√§hler metric (cscK metric), is (as the name suggests) a K√§hler metric on a complex manifold whose scalar curvature is constant. A special case is K√§hler‚ÄďEinstein metric, and a more general case is .

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  • Constant scalar curvature K√§hler metric
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  • In differential geometry, a constant scalar curvature K√§hler metric (cscK metric), is (as the name suggests) a K√§hler metric on a complex manifold whose scalar curvature is constant. A special case is K√§hler‚ÄďEinstein metric, and a more general case is .
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  • In differential geometry, a constant scalar curvature K√§hler metric (cscK metric), is (as the name suggests) a K√§hler metric on a complex manifold whose scalar curvature is constant. A special case is K√§hler‚ÄďEinstein metric, and a more general case is . , Tian and Yau conjectured that the existence of a cscK metric on a polarised projective manifold is equivalent to the polarised manifold being K-polystable. Recent developments in the field suggest that the correct equivalence may be to the polarised manifold being uniformly K-polystable. When the polarisation is given by the (anti)-canonical line bundle (i.e. in the case of Fano or Calabi‚ÄďYau manifolds) the notions of K-stability and K-polystability coincide, cscK metrics are precisely K√§hler-Einstein metrics and the Yau-Tian-Donaldson conjecture is known to hold.
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