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

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up vote 12 down vote accepted

Yes, see http://www.scholarpedia.org/article/Calabi-Yau_manifold#Moduli_of_high_dimensional_Calabi-Yau_manifolds

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thanks for the reference. – J Verma Jan 4 '11 at 21:50

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.

See here

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