  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.

Algebraic Geometry;Complex Variables