Let $B$ be a tropical affine manifold.

1. (1)

   Denote by $\Lambda\subseteq\mathcal{T}_{B}$ the local system of lattices generated locally by $\partial/\partial y_{1},\ldots,\partial/\partial y_{n}$, where $y_{1},\ldots,y_{n}$ are local affine coordinates. This is well-defined because transition maps are in $\mathbb{R}^{n}\rtimes GL_{n}(\mathbb{Z})$. Set

   $$X(B):=\mathcal{T}_{B}/\Lambda.$$

   This is a torus bundle over $B$. In addition, $X(B)$ carries a complex structure defined locally as follows. Let $U\subseteq B$ be an open set with affine coordinates $y_{1},\ldots,y_{n}$, so $\mathcal{T}_{U}$ has coordinate functions $y_{1},\ldots,y_{n}$, $x_{1}=dy_{1},\ldots,x_{n}=dy_{n}$. Then

   $$q_{j}=e^{2\pi i(x_{j}+iy_{j})}$$

   gives a system of holomorphic coordinates on $T_{U}/\Lambda|_{U}$, and the induced complex structure is independent of the choice of affine coordinates. This is called the semi-flat complex structure on $X(B)$.

   Later we will need a variant of this: for $\epsilon>0$, set

   $$X_{\epsilon}(B):=\mathcal{T}_{B}/\epsilon\Lambda.$$

   This has a complex structure with coordinates given by

   $$q_{j}=e^{2\pi i(x_{j}+iy_{j})/\epsilon}.$$

   (As we shall see later, the limit $\epsilon\rightarrow 0$ corresponds to a “large complex structure limit.”)

2. (2)

   Define $\check{\Lambda}\subseteq\mathcal{T}^{*}_{B}$ to be the local system of lattices generated locally by $dy_{1},\ldots,dy_{n}$, with $y_{1},\ldots,y_{n}$ local affine coordinates. Set

   $$\check{X}(B):=\mathcal{T}^{*}_{B}/\check{\Lambda}.$$

   Of course $\mathcal{T}^{*}_{B}$ carries a canonical symplectic structure, and this symplectic structure descends to $\check{X}(B)$.

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