Math. Ann. 245, 117 150 (1979) Am © by Springer-Verlag 1979 Toroidal Degeneration of Abelian Varieties. II* Yukihiko Namikawa** Department of Mathematics, Nagoya University, Furocho Chikusa-ku, Nagoya, Japan In this article and the ensuing we consider the following problem: Problem. Let X be an analytic space and U an analytic open subset of X (i.e. X - U is a closed analytic subset). For a 9iven smooth family of polarised abelian varieties 7t:d--*U, extend it to a family ~:d~X over X and study its properties. This problem was first studied extensively by Kodaira [5] in case fibres are elliptic curves and dimX = 1. In higher dimensional cases, since no theory of minimal models exists, there are many possibilities of extension. Mumford [8] gave a method of extension. Here we develop another more general method of constructing degenerate fibres by using the theory of toroidal embeddings due to Mumford et al. [-4] under two assumptions which are essential but not restrictive. The first is that (X, U) is a toroidal embedding (see Sect. 1 for the precise definition). Typical examples are those pairs (X, U) with X being smooth and X-U having only normal crossings with non-singular irreducible components. Thanks to Hironaka's theorem of resolution of singularieties we can always modify (X, U) so as to satisfy the above condition. The second assumption is that the monodromy (of H 1) is unipotent. Thanks to the theorem of quasiunipotentness of monodromy this assumption is fulfilled by taking a suitable ramified covering of X. (The general case without this assumption can be treated with this reduction, which we exhibit in the next part of this article.) In order to construct degenerate fibres over X- U we introduce a notion of"a set of degeneration data" (4.5). Our main theorem in Sect. 5 asserts that, given a set of degeneration data, then we can construct an extension of re. Several geometric properties of rc and fibres can be stated in terms of those of degeneration data (Sect. 5). Especially again (~', d) is a toroidal embedding and all fibres of ~ are also toroidal, hence we call this degeneration toroidal. In this article we put one more assumption that rc has a section o : U~d, and the general case is treated in the next part together with theories of degenerate theta functions and N6ron models. The above main theorem and other results were announced in the author's former article [11] under more restrictions. Generalisation here was done in three * Dedicated to Professor F. Hirzebruch on his 50th birthday ** Partially supported by Sonderforschungsbereich 40, "Theoretische Mathdmatik" 0025-5831/79/0245/0117/$06.80