The structure of Gauss-like maps. (English) Zbl 0547.14004

This paper studies a notion which generalizes the classical one of Gauss map attached to an immersion into projective space or an abelian variety. It is proved that such a generalized Gauss map, if proper, always Stein- factorizes through a holomorphically locally trivial fibration with homogeneous-space fiber. Special cases: an (immersed) submanifold of general type X in an abelian variety has finite Gauss map; equivalently, the canonical bundle \(K_ X\) is ample (conjecture of Ueno); an (immersed) nonlinear submanifold of projective space has finite Gauss map (first proved by Zak). A number of variants and other special cases are also presented.


14E25 Embeddings in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14K15 Arithmetic ground fields for abelian varieties
Full Text: Numdam EuDML


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