Asymptotic curvature of moduli spaces for Calabi-Yau threefolds

Understand

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.