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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.

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.

Reviewer: G.Van Steen (Antwerpen)

### MSC:

14K15 | Arithmetic ground fields for abelian varieties |

14G20 | Local ground fields in algebraic geometry |