zbMATH — the first resource for mathematics

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
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.