20 1. THE TROPICS One could, for example, study tropical curves inside tropical hypersurfaces, as has been done, say, in [111]. However, this is not the point of view we want to take here. Instead, we want to look at target spaces which “locally look like MR,” in such a way that we can still talk about tropical curves. The main point is that to talk about tropical curves, one needs the structure of MR as an affine space, but one also needs to know about the integral structure M MR. In what follows, we consider the group Aff(MR) = MR GL(MR) of affine linear automorphisms of MR, given by m Am + b, where A GLn(R) and b MR, and its subgroups MR GL(M) Aff(MR) and Aff(M) := M GL(M) Aff(MR). Definition 1.22. A tropical affine manifold is a real topological manifold B (possibly with boundary) with an atlas of coordinate charts ψi : Ui MR with transition functions ψi ψ−1 j MR GL(M) Aff(MR). An integral affine manifold is a tropical manifold with transition functions in Aff(M). We will often make use of two local systems on a tropical affine manifold, defining Λ TB to be the family of lattices locally generated by ∂/∂y1,...,∂/∂yn for y1,...,yn local affine coordinates. The sheaf ˇ T B is the dual local system locally generated by dy1,...,dyn. The point is these families of lattices are well- defined on tropical manifolds because of the restriction on the transition maps. Note that TB carries a natural flat connection, ∇B, with flat sections being R- linear combinations of ∂/∂y1,...,∂/∂yn. It is easy to generalize the notion of parameterized tropical curve with target a tropical affine manifold, as locally the notion of a line segment with rational slope and the balancing condition make sense. Example 1.23. B = MR/Γ for a lattice Γ MR gives an example of a compact tropical affine manifold. In [82], such spaces arise naturally as tropical Jacobians of tropical curves. Figure 15 gives an example of a two-dimensional torus containing a genus 2 tropical curve. Unfortunately, tori are not particularly useful for the applications we have in mind, so we shall generalize the notion of tropical affine manifold as follows. Definition 1.24. A tropical affine manifold with singularities is a topological manifold B along with data a subset Δ B which is a locally finite union of codimension 2 locally closed submanifolds of B a tropical affine structure on B0 := B \ Δ. We say B is an integral affine manifold with singularities if the affine structure on B0 is integral. The set Δ is called the singular locus or discriminant locus of B. Note that we are assuming that B is still a topological manifold even at the singular points: the singularities lie in the affine structure in the sense that, in general, the affine structure cannot be extended across Δ. We shall see some examples later, but for the moment one can imagine a two-dimensional cone as an
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