The Calabi conjecture for compact Kähler manifolds

Let $(X,g,\omega)$ be a compact Kähler manifold, and fix $R\in\Omega^{1,1}(X)$ such that $[R]=c_1(T_X)\in H^{1,1}(X)$. Then there exists a unique Kähler metric $\widetilde{g}$ with Kähler form $\widetilde{\omega}$ such that $[\omega]=[\widetilde{\omega}]$ and

$$R=\operatorname{Ric}(\widetilde{\omega}),$$

where $\operatorname{Ric}(\widetilde{\omega})$ is the Ricci form of $\widetilde{\omega}$. The Calabi conjecture. The curvature data of a compact Kähler manifold is determined in this way by its first Chern class. Calabi proved uniqueness, while Yau later proved existence, so the conjecture is solved; its power is that it describes complicated geometric data in terms of simpler topological data.