K.C3.A4hler.E2.80.93Einstein_manifolds Kähler_manifold
a kähler manifold called kähler–einstein if has constant ricci curvature. equivalently, ricci curvature tensor equal constant λ times metric tensor, ric = λg. reference einstein comes general relativity, asserts in absence of mass spacetime 4-dimensional lorentzian manifold 0 ricci curvature. see article on einstein manifolds more details.
although ricci curvature defined riemannian manifold, plays special role in kähler geometry: ricci curvature of kähler manifold x can viewed real closed (1,1)-form represents c1(x) (the first chern class of tangent bundle) in h(x,r). follows compact kähler–einstein manifold x must have canonical bundle kx either anti-ample, homologically trivial, or ample, depending on whether einstein constant λ positive, zero, or negative. kähler manifolds of 3 types called fano, calabi–yau, or ample canonical bundle (which implies general type), respectively. kodaira embedding theorem, fano manifolds , manifolds ample canonical bundle automatically projective varieties.
yau proved calabi conjecture: every smooth projective variety ample canonical bundle has kähler–einstein metric (with constant negative ricci curvature), , every calabi–yau manifold has kähler–einstein metric (with 0 ricci curvature). these results important classification of algebraic varieties, applications such miyaoka–yau inequality varieties ample canonical bundle , beauville–bogomolov decomposition calabi–yau manifolds.
by contrast, not every smooth fano variety has kähler–einstein metric (which have constant positive ricci curvature). however, chen–donaldson–sun proved yau–tian–donaldson conjecture: smooth fano variety has kähler–einstein metric if , if k-stable, purely algebro-geometric condition.
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