Families over curves with a strictly maximal Higgs field

Abstract

We study certain rigid Shimura curves in the moduli scheme of polarized minimal n-folds of Kodaira dimension zero. Those are characterized by some numerical condition on the Deligne extension of the corresponding variation of Hodge structures, or equivalently by the strict maximality of the induced Higgs field. We show that such Shimura curves can not be proper for n odd, and we give some examples, showing that they exist in all dimensions. Let f : X → Y be a family of n-dimensional complex algebraic varieties, smooth over U = Y \S. We will assume that X is projective and non-singular, that Y is a smooth projective curve, and that the general fibre F of f is connected, hence irreducible. Writing X0 = f −1(U) one has the Q variations of Hodge structures Rf∗QX0 of weight k. Usually we will assume that the monodromy around each point s ∈ S is unipotent and that the family is not birationally isotrivial. Considered the Deligne extension of (Rf∗QX0) ⊗ OU to Y together with the extension of the Hodge filtration. Taking the graded sheaf one obtains the Higgs bundle (E, θ) = ( ⊕