24 1. THE TROPICS If furthermore all polyhedra in P are lattice polytopes, then in fact the aﬃne structure is integral. This construction provides a wide class of examples. However, these examples are still too general. We will impose one additional condition. We say a collection of fan structures {Sv | v ∈ P[0]} is compatible if, for any two vertices v, w of τ ∈ P, the fan structures induced on τ by Sv and Sw are equivalent. Note that given such a compatible set of fan structures, we obtain well-defined fan structures along every τ ∈ P.3 Definition 1.27. A tropical manifold is a pair (B, P) where B is a tropical aﬃne manifold with singularities obtained from the polyhedral decomposition P of B and a compatible collection {Sv | v ∈ P[0]} of fan structures. (B, P) is an integral tropical manifold if in addition all polyhedra in P are lattice polyhedra. Examples 1.28. (1) Any lattice polyhedron σ with at least one vertex supplies an example of an integral tropical manifold, either bounded or unbounded, with B = σ and P the set of faces of σ. In this case the aﬃne structure on Int(σ) extends to give the structure of an aﬃne manifold (with boundary) on σ. Here Δ = ∅. (2) The polyhedral decomposition of B = MR given in Definition 1.4 is also an example of a tropical manifold (provided the tropical hypersurface in question has at least one vertex). (3) Let Ξ ⊆ MR be a reflexive lattice polytope, i.e., 0 ∈ Int(Ξ) and the polytope Ξ∗ := {n ∈ NR | n, m≥ −1 ∀m ∈ Ξ} is also a lattice polytope. Then B = ∂Ξ carries the obvious polyhedral decomposition P consisting of the proper faces of Ξ. These faces are lattice polytopes. So, to specify an integral tropical manifold structure on B, we need only specify a fan structure at each vertex v of Ξ. This is done via the projection Sv : Uv → MR/Rv. Compatibility is easily checked, as the induced fan structure on a cell ω ∈ P containing v is the projection Uω → MR/Rω, where Rω now denotes the vector subspace of MR spanned by ω. There are a number of refinements of this construction. For example, if we take a refinement P of P by integral lattice polytopes, we can use the same prescription above for the fan structure at the vertices. Note that, in this case, the description of Δ ⊆ B of the singular locus as determined by the refinement P may give a much bigger discriminant locus, with Δ ∩ Int(σ) = ∅ for σ a maximal proper face of Ξ. However, the aﬃne structure induced by P on Int(σ) is compatible with the obvious aﬃne structure on Int(σ), so in fact the aﬃne structure extends across points of Δ ∩ Int(σ). Thus we can replace Δ with Δ ∩ τ∈P dim τ=dim Ξ−2 τ. For example, let Ξ1 := Conv{(−1, −1, −1), (3, −1, −1), (−1, 3, −1), (−1, −1, 3)}. 3 In the case we will focus on in this book, dim B = 2, this compatibility condition is in fact trivial, since, provided v = w, τ is dimension one or two and there are not many choices for zero or one-dimensional fans! So, for the most part, the reader can ignore this condition.

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