Kähler-Einstein metrics with non-positive scalar curvature

Abstract

We have seen that the Ricci curvature represents the first Chern class. In this section, we will consider the converse problem, namely, given a Kähler class [ω] ∈ H² (M, ℝ) ∩ H^1,2 (M, ℂ) on a compact Kähler manifold M and any form Ω representing the first Chern class, can we find a metric ω ∈ [ω] such that Ric(ω) = Ω? This is known as the Calabi conjecture and it was solved by Yau in 1976. We will state it here as a theorem and refer to it as the Calabi-Yau Theorem.