ScalingStacks

The smooth case [02FL]

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The smooth case

The link between Monge-Ampère equations and Kähler-Einstein metrics is provided by the following classical

Lemma 7.1.

Let XX be complex manifold, let hh be a smooth hermitian metric on ωX\omega_{X} and Ω\Omega a Kähler form such that Ωn=v⁡(h)\Omega^{n}=v(h). The Ricci curvature divided by 2​π2\pi of Ω\Omega is the Chern-Weil form −c1​(KX,h)-c_{1}(K_{X},h).

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