The ideas of this paper were suggested by the formal likeness between Mumford's toric description of degenerating families of elliptic curves (in [A]) and parabolic Inoue surfaces (first constructed in [In]). The likeness is not readily apparent from [In] but is pointed out in [MO], where a toric construction of parabolic Inoue surfaces is given. The construction is, in a manner of speaking, dual to Mumford's.
Mumford's construction has been extended in many ways. The purpose of this paper is to show how one such extension may be dualised (in the loose sense just mentioned) so as to give a construction of some new complex analytic n-folds, especially 3-folds, and to study these manifolds.
The starting point is [FS-B], in which Mumford's ideas are used to construct degenerations of Kodaira surfaces. It is also possible to write down straightforward, but apparently also new, higher-dimensional analogues of Kodaira surfaces, and to apply the same constructions to them.
1. The three-fold case
In this section I shall describe the dualisation procedure as it applies to 1-parameter degenerations of Kodaira surfaces. The methods of toric geometry will be used freely: references are [MO], [O] and [Da].
Let N = Z3 c R3 be the standard lattice of integer points and let x,, x,, x3 be the standard coordinates. Let a,, a,, b be positive integers and let G be the subgroup of SL(N) generated by y1 and y,, where y l : ( x l ? x2, X3)H(X1, x2, x 3 + a l x l ) , y2:(x1, x2, X3)H(X1, x2 + u 2 x 1 , x3 + bx,).
G is thus free abelian of rank 2. Proceeding as in [FS-B], we take the 3-simplices generated by (1, r, s), (1, r + 1, s) and (1, r + 1, s + 1) and by (1, r, s), (1, r, s + 1) and (1, r + 1, s + 1) for 0 ,< r < a,, 0 < s < a,, r, s E Z. These form a basic subdivision of the cone 6'