Intersection complexes of fans and toric varieties

In [GM2], Goresky and MacPherson defined and constructed intersection complexes for topological pseudomanifolds. The complexes are defined in the derived category of sheaves of modules over a constant ring sheaf. Since analytic spaces are of this category, algebraic varieties defined over C have intersection complexes. The intersection complex of a given variety has a variation depending on a sequence of integers which is called a perversity [GM2, §2]. It is known that the complex with the middle perversity is the most important for normal complex varieties. In particular, the decomposition theorem and the strong Lefschetz theorem for the intersection cohomologies with respect to the middle perversity were proved in [BBD]. On the other hand, toric varieties are known to be described by rational fans in a real space with a lattice. By applying the strong Lefschetz theorem for toric varieties, Stanley proved the g-theorem which was a combinatorial conjecture on the number of faces of a simplicial convex polytope [S2]. As an attempt to prove the g-theorem by a combinatorial method, Oda defined complexes of real vector spaces for not necessarily rational fans in a real space [O3]. He reduced a weak assertion of the Lefschetz theorem for a toric variety to the vanishing of some cohomologies of these complexes associated to the corresponding rational fan. In this article, we complete Oda’s plan for the combinatorial proof of g-conjecture by proving the vanishing of cohomologies in a general form. Furthermore, we show the unimodality of the generalized h-vector of a not necessarily rational fan if it has a strongly convex piecewise linear function (cf. Theorems 7.6 and 7.7). Since the fan associated to