ScalingStacks

We now give a very sketchy account of (the modern view on) Yau’s proof of Theorem 4.1. Fix a background Kähler metric $\omega$ in the given class, and the task is to find a Kähler potential $\phi$ solving the complex Monge-Ampère equation for any given density function $e^{f}$ satisfying the cohomological constraint $\int_{X}e^{f}\omega^{n}=\int_{X}\omega^{n}$,

$$\omega_{\phi}^{n}=(\omega+dd^{c}\phi)^{n}=e^{f}\omega^{n},\quad\int_{X}\phi\omega^{n}=0,$$

(9)

In particular for $e^{f}\omega^{n}=\text{const}\cdot\Omega\wedge\overline{\Omega}$ one obtains the Calabi-Yau metric. The uniqueness follows from a simple integration by parts argument.