We now describe some aspects of the work of Wei-Dong Ruan in [18, 19, 20, 21]. This is based on the following idea. Suppose we are given a family of Calabi–Yau hypersurfaces $X$ in some projective toric variety. As in [18, 19, 21] we take this to be a pencil of quintics $\bigl\{X_{c}:c\in\mathbin{\mathbb{C}}\cup\{\infty\}\bigr\}$ in $\mathbb{CP}^{4}$, where

$$X_{c}=\bigl\{[z_{0},\ldots,z_{4}]\in\mathbb{CP}^{4}:p(z_{0},\ldots,z_{4})+c\,q(z_{0},\ldots,z_{4})=0\bigr\},$$

and $p,q$ are homogeneous, linearly independent quintic polynomials.

Choose a Kähler metric $g$ on $\mathbb{CP}^{4}$, with Kähler form $\omega$. Let $s$ be the meromorphic function $p(z_{0},\ldots,z_{4})/q(z_{0},\ldots,z_{4})$ on $\mathbb{CP}^{4}\setminus X_{\infty}$, and let $f=\mathop{\rm Re}(s)$. Define a vector field $v$ on $\mathbb{CP}^{4}$ by $v^{a}=|{\rm d}f|^{-2}g^{ab}({\rm d}f)_{b}$, using the index notation for tensors. Note that $v$ becomes infinite on $X_{\infty}$, as $f$ is infinite there, and also on the set of points where ${\rm d}f=0$. Ruan shows that flowing along the vector field $v$ for time $t$ takes $X_{c}$ to $X_{c+t}$ for each $c\in\mathbin{\mathbb{C}}$, at least where $v$ is finite. Furthermore, the flow takes Lagrangian submanifolds of $X_{c}$ to Lagrangian submanifolds of $X_{c+t}$.

Ruan’s method is to set $p(z_{0},\ldots,z_{4})=z_{0}z_{1}z_{2}z_{3}z_{4}$, so that $X_{0}$ is the union of five copies of $\mathbb{CP}^{3}$ in $\mathbb{CP}^{4}$, a very degenerate, singular quintic. He defines an explicit Lagrangian fibration of $X_{0}$, with respect to the Fubini–Study metric on $\mathbb{CP}^{4}$. Then he uses the flow from $X_{0}$ to $X_{t}$ to translate this fibration to a Lagrangian fibration of the general, nonsingular quintic $X_{t}$ for $t\in\mathbin{\mathbb{R}}\setminus\{0\}$. One has to consider carefully what happens when $v$ is infinite, and around the singularities of $X_{0}$. But it turns out that these do not spoil things, and we end up with a genuine Lagrangian fibration of $X_{t}$.

Part of the motivation for Ruan’s construction is that $X_{0}$ is considered to be the ‘large complex structure limit’ of Calabi–Yau quintics. Thus, the construction starts with an explicit fibration of the singular ‘large complex structure limit’ 3-fold, and deforms it to a fibration of nonsingular 3-folds close to this limit. This is quite a natural thing to do from the String Theory point of view, and others such as Zharkov and Goldstein have tried similar ideas.

Now we are interested in the nature of the set of singular fibres in Ruan’s fibrations, and in their singularities. Ruan proves [19, Th. 2.2]:

The properties of Ruan’s fibrations given above are very similar to the fibrations we shall propose later in the paper. In particular, versions of parts (i) and (iii)–(vi) will hold for our fibrations. For part (ii), the set of singular points in the fibrations we shall discuss need not be a holomorphic curve, but it will be a real 2-manifold in $X$ that is close to being holomorphic. Only in part (vii) do we seriously diverge from Ruan, as our fibrations will not contain fibres in which $T^{2}$ collapses to a point.

Ruan himself, however, appears to regard these properties of his fibrations as a problem (see for instance [21, Conj. 1.1], where he conjectures that special Lagrangian fibrations are always smooth, the ‘Precise SYZ mirror conjecture’ in [20, §9], and many other places), and spends much effort in showing how to deform his fibrations to smooth Lagrangian fibrations. One moral of this paper may be that Ruan’s construction gives something quite close to the right answer, and it might even be possible to modify it to yield genuine special Lagrangian fibrations of (almost) Calabi–Yau manifolds.