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Motivated by the classical statements of Mirror Symmetry, we study certainKahler metrics on the complexified Kahler cone of a Calabi-Yau threefold,conjecturally corresponding to approximations to the Weil-Petersson metric nearlarge complex structure limit for the mirror. In particular, the naturallydefined Riemannian metric (defined via cup-product) on a level set of theKahler cone is seen to be analogous to a slice of the Weil-Petersson metricnear large complex limit. This enables us to give counterexamples to aconjecture of Ooguri and Vafa that the Weil-Petersson metric has non-positivescalar curvature in some neighbourhood of the large complex structure limitpoint.

Asymptotic curvature of moduli spaces for Calabi-Yau threefolds