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Conjecture of Zhiqin Lu : Calabi-Yau moduli space is a concave manifold?.

Definition: By concavity, we mean that there exists an exhaustion function on the manifold such that at each point the (complex) Hessian form has at least two negative eigenvalues.

Is there any progress for his conjecture?

Moreover , moduli space of Fano Kahler-Einstein manifolds is concave? Or what about concavity of the moduli space of general type manifolds when their Canonical bundle is ample?

The Study of concavity of moduli spaces corresponds to Weil-Petersson geometry.

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