Deformation of pairs and semiregularity

We study relative deformation of a map into a Kähler manifold whose image is a divisor.We show that if the map satisfies a condition called semiregularity, then it allows relative deformations if and only if the cycle class of the image remains Hodge in the family.This gives a refinement of the so-called variational Hodge conjecture.We also show that the semiregularity of maps is related to classical notions such as Cayley-Bacharach conditions and d-semistability.