ScalingStacks

A real $n$-dimensional submanifold $L$ of a compact Calabi-Yau n-fold $(X,\omega,J,\Omega)$ is called a special Lagrangian (SLag) with phase angle $\theta$ if

$$\omega|_{L}=0,\quad\text{Im}(e^{i\theta}\Omega)|_{L}=0.$$

(6)

They are special cases of calibrated submanifolds introduced by Harvey and Lawson [25], and in particular are minimal submanifolds. The classical result of McLean says that the deformation theory of SLags with phase $\theta$ is unobstructed, and the first order deformation space is isomorphic to $H^{1}(L,\mathbb{R})$. Thus if $L$ is diffeomorphic to $T^{n}$, then the deformation space is $n$-dimensional, compatible with the SYZ conjecture that $X$ admits a SLag $T^{n}$-fibration. A sufficient condition to construct Slag fibrations, under the very strong hypothesis of collapsing metric with locally bounded sectional curvature, is obtained by Zhang [41, Thm 1.1].

The essence of Zhang’s result is a standard application of the implicit function theorem, and we shall summarize the key points (cf. [41, section 4] for more details). Denote $Y_{r}=T^{n}\times B(0,r)\subset T^{n}_{x_{i}}\times\mathbb{R}^{n}_{y_{i}}\simeq T^{*}T^{n}$, where $r\gg 1$ is fixed. The trivial example of a SLag fibration is the following: the CY structure is the flat model

$$g=\sum(dx_{i}^{2}+dy_{i}^{2}),\quad\omega=\sum dx_{i}\wedge dy_{i},\quad\Omega=\bigwedge(dx_{j}+\sqrt{-1}dy_{j}),$$

and the Slag fibration is just the projection to the $\mathbb{R}^{n}_{y_{i}}$ factor, namely the tori $T^{n}\times\{y\}$ are SLags. Zhang considers a family of CY structures $(g_{k},\omega_{k},\Omega_{k})$ converging to $(g,\omega,\Omega)$ in the $C^{\infty}$-sense on $Y_{2r}$ (which follows from his bounded sectional curvature assumptions by elliptic bootstrap), such that $\omega_{k}\in[\omega]\in H^{2}(Y_{2r},\mathbb{R})$. Small deformations of the standard $T^{n}$ fibres can be represented as graphs on $T^{n}$: for $y\in\mathbb{R}^{n}$ and a 1-form $\sigma$ on $T^{n}$ orthogonal to the harmonic 1-forms $dx_{1},\ldots dx_{n}$, write

$$L(y,\sigma)=\text{Graph}(x\mapsto y+\sigma(x))\subset T^{*}T^{n}.$$

The condition for $L(y,\sigma)$ to be a SLag with respect to $(g_{k},\omega_{k},\Omega_{k})$ is

$$\omega_{k}|_{L(y,\sigma)}=0,\quad\text{Im}(e^{\sqrt{-1}\theta_{k}}\Omega_{k})|_{L(y,\sigma)}=0,$$

(7)

where $\theta_{k}$ are chosen so that $\int_{T^{n}}e^{\sqrt{-1}\theta_{k}}\Omega_{k}>0$. Zhang shows by perturbation arguments that for each $y\in B(0,\frac{3r}{2})$ and $k\geq k_{0}\gg 1$, there is a unique $\sigma=\sigma_{k,y}$ such that $L(y,\sigma_{k,y})$ solves (7) with small norm bound $\left\lVert\sigma_{k,y}\right\rVert<\delta\ll 1$. He then uses another implicit function argument to show that these SLags indeed define a local SLag $T^{n}$-fibration on some open subset of $Y_{3r/2}$ containing $Y_{r}$.