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Degenerating abelian varieties. (English) Zbl 0761.14015
The paper deals with abelian varieties over a field $$K$$ which is complete with respect to a valuation of height 1. Certain statements which were formulated by Raynaud in 1970 are proved.
In a first part the authors give an overview of results on uniformizations of abelian varieties in the framework of rigid analytic geometry. It turns out that after a suitable base extension an abelian variety over $$K$$ has a uniformization $$E$$ which is an extension of an abelian variety with good reduction $$(B)$$ by a split torus $$(T)$$. The abelian variety is the quotient $$E/M$$ where $$M$$ is a lattice in $$E$$ with rank $$\dim(T)$$. — Starting with an extension sequence $$0\to T\to E\to B\to 0$$ and a lattice $$M$$ in $$E$$ of rank $$\dim(T)$$ the quotient $$E/M$$ exists as an analytic group and $$E/M$$ is an abelian variety if and only if certain Riemann conditions are satisfied. The formulation of these Riemann conditions is given and the equivalence is proved, first in the case that $$B$$ is trivial and later in the general case. — Furthermore dual varieties are constructed and polarizations are studied, both in the case that $$B$$ is trivial and in the general case. — The paper concludes with a short section which relates the results with the approach of Mumford, Faltings and Chai.

##### MSC:
 14K15 Arithmetic ground fields for abelian varieties 14G20 Local ground fields in algebraic geometry
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