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Special families of curves, of Abelian varieties, and of certain minimal manifolds over curves

Research paper by Martin Moeller, Eckart Viehweg, Kang Zuo

Indexed on: 07 Dec '05Published on: 07 Dec '05Published in: Mathematics - Algebraic Geometry



Abstract

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