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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.

MSC:
14E25 Embeddings in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14K15 Arithmetic ground fields for abelian varieties
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References:
[1] W. Fulton and R. Lazarsfeld : Connectivity and its applications in algebraic geometry . Lecture Notes 862, pp. 26-92. · Zbl 0484.14005
[2] P. Griffiths and J. Harris : Algebraic geometry and local differential geometry . Ann. Scient. Ec. Norm. Sup (4) 12 (1979) 355-432. · Zbl 0426.14019 · doi:10.24033/asens.1370 · numdam:ASENS_1979_4_12_3_355_0 · eudml:82039
[3] K. Ueno : Classification theory of algebraic varieties and compact complex spaces . Lecture notes 439. Berlin: Springer 1975. · Zbl 0299.14007
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