is complex moduli space of a Calabi - Yau Kahler

The complex moduli space of a Calabi-Yau manifold is a complex manifold (BTT). Is it also Kahler ?

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Let $\mathcal M$ be a complex moduli space of a Calabi-Yau manifolds, then there is a symplectic metric on $\mathcal M$ which is known as Weil Petersson metric and Gang Tian showed that it is Kahler and later Georg Schomacher extended it for pair $(X,D)$ where $D$ is a divisor and proved that logarithmic Weil petersson metric on Log Fano varieties is also Kahler.