General infinitesimal variations of Hodge structure of ample curves in surfaces

Given a smooth projective complex curve inside a smooth projective surface, one can ask how its Hodge structure varies when the curve moves inside the surface. In this paper we develop a general theory to study the infinitesimal version of this question in the case of ample curves. We can then apply the machinery to show that the infinitesimal variation of Hodge structure of a general deformation of an ample curve in ${\mathbb{P}}^1\times {\mathbb{P}}^1$ is an isomorphism.