Ranks of Kodaira-Spencer classes and Clifford indices

Given a family of compact Kähler manifolds, the infinitesimal behaviour of the associated period maps (the Infinitesimal Torelli problem) depends on some cup-products with the Kodaira-Spencer class of the deformation. For families of curves, it was noted by Ginensky that the rank of such cup-product homomorphisms can be bounded below by the Clifford Index of a suitable divisor. In this talk I will present some work in progress, attempting to generalize Ginensky's result to families of higher-dimensional manifolds.