Not long after its discovery, mirror symmetry has been extended to the non–Calabi-Yau setting, notably to Fano manifolds, through the works of Batyrev [10], Givental [58, 59, 60], Kontsevich [103], Hori-Vafa [89] and many others. Unlike the Calabi-Yau case, the mirror is no longer given by a manifold; instead, it is predicted to be a pair $(\check{X},W)$, where $\check{X}$ is a non-compact Kähler manifold and $W:\check{X}\to\mathbb{C}$ is a holomorphic function. In the physics literature, such a pair $(\check{X},W)$ is called a Landau-Ginzburg model, and $W$ is called the superpotential of the model [159, 167].

It is natural to ask whether the SYZ proposal continues to work in this setting as well. Auroux [8] was the first to consider this question and in fact he extended the SYZ proposal to a much more general setting. Namely, he considered pairs $(X,D)$ consisting of a compact Kähler manifold $X$ together with an effective anticanonical divisor $D$. The defining section of $D$ gives a holomorphic volume form on $X\setminus D$ with simple poles along the divisor $D$, so it makes sense to speak about special Lagrangian torus fibrations on the complement $X\setminus D$. Suppose that we are given such a fibration $\mu:X\setminus D\to B$, then we can try to produce the SYZ mirror $\check{X}$ by $T$-duality (i.e. consider the moduli space of pairs $(L,\nabla)$ where $L$ is a fiber of $\mu$ and $\nabla$ is a flat $U(1)$-connection over $L$) modified by instanton corrections. Moreover, the superpotential $W$ will naturally appears as the object mirror to Fukaya-Oh-Ohta-Ono’s obstruction chain $\mathfrak{m}_{0}$.

When $X$ is a compact toric Kähler manifold, a canonical choice of $D$ is the union of all toric prime divisors. Also, the moment map provides a convenient Lagrangian torus fibration on $X$, which has the nice property that it restricts to a torus bundle on the open dense torus orbit $X\setminus D$. In this case, the SYZ mirror manifold $\check{X}$ is simply given by the algebraic torus $(\mathbb{C}^{\times})^{n}$, because we have a torus bundle and there are no instanton corrections in the construction of the mirror manifold. All the essential information is encoded in the superpotential $W$. Prior to the work of Auroux, it was Cho and Oh [35, 37] who first noticed that $W$ can be expressed in terms disk counting invariants (or open Gromov-Witten invariants). By classifying all holomorphic disks in $X$ bounded by moment map fibers, they got an explicit formula for $W$ in the case when $X$ is Fano, and this agrees with earlier predictions obtained using physical arguments by Hori and Vafa [89]. This was later vastly generalized by the works of Fukaya, Oh, Ohta and Ono [54, 55, 52] on Lagrangian Floer theory and mirror symmetry for toric manifolds.

In [21], mirror symmetry for toric Fano manifolds was used as a testing ground to see how useful Fourier-Mukai–type transforms, or what we call SYZ transforms, could be in the investigation of the geometry of mirror symmetry. For a toric Fano manifold $X$, we consider the open dense torus orbit $X_{0}:=X\setminus D\subset X$, which is also the union of Lagrangian torus fibers of the moment map. Symplectically, we can write $X_{0}=T^{*}B_{0}/\Lambda^{\vee}$, where $B$ is the moment polytope and $B_{0}$ denotes its interior. Then the SYZ mirror is $\check{X}:=TB_{0}/\Lambda$ which is a bounded domain in $(\mathbb{C}^{\times})^{n}$. To obtain the superpotential $W$, we consider the space

$$\tilde{X}:=X_{0}\times\Lambda\subset\mathcal{L}X$$

of fiberwise geodesic/affine loops in $X$. On $\tilde{X}$, we have an instanton-corrected symplectic structure $\tilde{\omega}=\omega+\Phi$, where $\Phi$ is a generating function of genus 0 open Gromov-Witten invariants which count (virtually) holomorphic disks bounded by moment map fibers.

An explicit SYZ transform $\mathcal{F}$ was then constructed by combining the semi-flat SYZ transform $\mathcal{F}^{\textrm{semi-flat}}$ with fiberwise Fourier series, and it was shown that $\mathcal{F}$ transforms the corrected symplectic structure $\tilde{\omega}$ on $X$ precisely to the holomorphic volume form $e^{W}\check{\Omega}$ of the mirror Landau-Ginzburg model $(\check{X},W)$, where $W$ was obtained by taking fiberwise Fourier transform of $\Phi$. Moreover, $\mathcal{F}$ induces an isomorphism between the (small) quantum cohomology ring $QH^{*}(X)$ of $X$ and the Jacobian ring $Jac(W)$ of $W$. The proof was by passing to the tropical limit, and observing that a tropical curve whose holomorphic counterpart contributes to the quantum product can be obtained as a gluing of tropical disks (see the work [21] for more details). This observation was later generalized and used by Gross [70] in his study of mirror symmetry for the big quantum cohomology of $\mathbb{P}^{2}$ via tropical geometry.

As for manifolds of general type, there are currently two main approaches to their mirror symmetry along the SYZ perspective. One is the work of Abouzaid, Auroux and Katzarkov, where they considered a hypersurface $H$ in a toric variety $V$ and constructed a Landau-Ginzburg model which is SYZ mirror to the blowup of $V\times\mathbb{C}$ along $H\times\{0\}$. In particular, when $H$ is the zero set of a bidegree $(3,2)$ polynomial in $V=\mathbb{P}^{1}\times\mathbb{P}^{1}$, their construction produces a mirror of the genus 2 Riemann surface, which is in agreement with a previous proposal by Katzarkov [99, 97, 146].

Another approach, which is more in line with the Gross-Siebert program, is the work by Gross, Katzarkov and Ruddat [73]. They proposed that the mirror to a variety of general type is a reducible variety equipped with a certain sheaf of vanishing cycles. Presumably, the mirror produced in this approach should give the same data as the one produced by [3]. For example, the reducible variety should the critical locus of the superpotential of the SYZ mirror Landau-Ginzburg model. But the precise relations between these two approaches are still under investigation.