1.4. AFFINE MANIFOLDS WITH SINGULARITIES 23 If : Rk is a fan structure along τ P and σ τ, then Uτ. We then obtain a fan structure along σ induced by given by the composition Uτ−→Rk Rk/Lσ Rl where Rk is the linear span of Kτ,σ. This is well-defined up to equivalence. It is easy to see that with the induced fan structure on σ, Σσ = Στ(Kτ,σ) in the notation of Definition 1.10. See Figure 16 for a picture of a fan structure. The most important case is when τ = v a vertex of P. Then a fan structure is an identification of a neighbourhood of v in B with a neighbourhood of the origin in Rn (n = dim B). This identification locally near v identifies P with a fan Σv in Rn. Given a fan structure Sv at each vertex v P, we can construct a tropical structure on B as follows. We first need to choose a discriminant locus Δ B. The precise details of this choice of discriminant locus in fact turn out not to be so important, and it can be chosen fairly arbitrarily, subject to certain constraints: (1) Δ does not contain any vertex of P. (2) Δ is disjoint from the interior of any maximal cell of P. (3) For any ρ P which is a codimension one cell not contained in ∂B, the connected components of ρ \ Δ are in one-to-one correspondence with vertices of ρ, with each vertex contained in the corresponding connected component. For example, if dim B = 2, we simply choose one point in the interior of each compact edge of P not contained in ∂B, and take Δ to be the set of these chosen points. If B is compact without boundary of any dimension, we can take Δ to be the union of all simplices in the first barycentric subdivision of P which neither contain a vertex of P nor intersect the interior of a maximal cell of P. For the general case, see [49], §1.1. We should also note that, for us, this is a maximal choice of discriminant locus, and if there is a subset Δ Δ such that the affine structure on B \ Δ extends to an affine structure across B \ Δ , we will replace Δ with Δ without comment. For a vertex v of P, let Wv denote a choice of open neighbourhood of v with Wv Uv satisfying the condition that if v ρ with ρ a codimension one cell, then Wv ρ is the connected component of ρ \ Δ containing v. Then {Int(σ) | σ Pmax} {Wv | v P[0]} form an open cover of B0 := B \ Δ. We define an affine structure on B0 via the already given affine structure on Int(σ) for σ Pmax, ψσ : Int(σ) MR and the composed maps ψv : Wv Uv−→Rdim Sv B where the first map is the inclusion. It is easy to see that this produces the structure of a tropical affine manifold with singularities on B. Indeed, the crucial point is that the affine charts ψv induced by the choice of fan structure are compatible with the charts ψσ on the interior of maximal cells of P, but this follows precisely from item (2) in the definition of a fan structure.
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