In the generic region $f:M\to B$ is a smooth proper submersion with $T^{3}$ fibres. A torus fibration is called semiflat if the metric restricts to flat metrics on the tori. The Calabi-Yau structure $(g,\omega,\Omega)$ on a semiflat SYZ $T^{3}$-fibration can be locally described in action-angle coordinates as

$$\begin{cases}\omega=\sum_{1}^{3}d\mu_{i}\wedge d\theta_{i},\\ \Omega=(\det(g_{ij}))^{-1/2}\bigwedge_{i=1}^{3}(d\theta_{i}-\sqrt{-1}g_{ij}d\mu_{j}),\\ g=g^{ij}d\theta_{i}d\theta_{j}+g_{ij}d\mu_{i}d\mu_{j}.\end{cases}$$

Here $g_{ij}$ is the Hessian of a real valued function $\varphi$ on $B$ solving the real Monge-Ampère equation:

$$g_{ij}=\frac{\partial^{2}\varphi}{\partial\mu_{i}\partial\mu_{j}},\quad\det(\frac{\partial^{2}\varphi}{\partial\mu_{i}\partial\mu_{j}})=\text{const}.$$

and $g_{ij}$ defines a metric on the base $g_{B}=g_{ij}d\mu_{i}d\mu_{j}$ such that $f:M\to B$ is a Riemannian submersion.

The Calabi-Yau structure induce two sets of affine structures on the base $B$: the symplectic moment coordinates $\mu_{i}$ satisfying $d\mu_{i}=-\omega(\frac{\partial}{\partial\theta_{i}},\cdot)$, and the complex affine coordinates $y_{i}$ satisfying $dy_{i}=\text{Im}\Omega(\frac{\partial}{\partial\theta_{j}},\frac{\partial}{\partial\theta_{k}},\cdot)$ for cyclic indices $i,j,k$. These coordinates are related by the Legendre transform $y_{i}=-\frac{\partial\varphi}{\partial\mu_{i}}.$

Semiflat mirror symmetry is the observation that if over the same base $B$ we fibrewise replace $T^{3}$ by the dual tori, then there is a canonical Calabi-Yau structure exhibiting this dual torus fibration as a semiflat SYZ fibration. Furthermore, the base metric $g_{B}$ is unchanged while the roles of the two affine structures are interchanged.

A major part of the SYZ Conjecture 1.1 is that Calabi-Yau metrics on (polarised) manifolds near the large complex structure limit are asymptotically described by such semiflat SYZ fibrations in the generic region up to exponentially small errors.