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