Sectional curvatures of Kahler moduli

Abstract

We investigate a new property for compact Kahler manifolds. Let X be a Kahler manifold of dimension n and let H^{1,1} denote the (1,1) part of its real second cohomology. On this space, we have an degree n form given by cup product. Let K denote the open cone of Kahler classes in H^{1,1}, and K_1 the level set consisting of classes in K on which the n-form takes value one. This is a Riemannian manifold, with tangent space at a given point being the primitive classes of type (1,1), and metric defined via the Hodge Index Theorem. In the Calabi-Yau case (and probably more generally), we conjecture that K_1 has non-positive sectional curvatures. This would place new restrictions on the possible location of the Kahler cone in cohomology, giving potentially useful information as to which differentiable manifolds may support Calabi-Yau structures. The conjecture is motivated by a Mirror Symmetry argument in Section 1. This argument suggests that one should develop a mirror version of the Weil-Petersson theory of complex moduli. The outline of such a theory is described in Sections 2-4, and the conjecture is verified under certain extra assumptions. In Section 5, we investigate in more detail the case when X is a Kahler threefold with h^{1,1} = 3, where we only have one sectional curvature on K_1 to consider. We prove a formula (5.1) relating this curvature to the classical invariants of the ternary cubic form, and we discuss various implications of this formula.