Abstract
 

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I will report a recent joint work with Eckart Viehweg. 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 dimension