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

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2011 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.