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This survey article discusses some results on the structure of familiesf:V-->U of n-dimensional manifolds over quasi-projective curves U, withsemistable reduction over a compactification Y of U. We improve the Arakelovinequality for the direct images of powers of the dualizing sheaf. For familiesof Abelian varieties we recall the characterization of Shimura curves byArakelov equalities. For families of curves we recall the characterization ofTeichmueller curves in terms of the existence of certain sub variation of Hodgestructures. We sketch the proof that the moduli scheme of curves of genus g>1can not contain compact Shimura curves, and that it only contains a non-compactShimura curve for g=3.

Special Families of Curves, of Abelian Varieties, and of Certain Minimal Manifolds over Curves