7.1 A conjectural local model for SL fibrations [03M2]

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7.1 A conjectural local model for SL fibrations

We shall proceed by giving a series of assumptions that define the properties of the fibrations we seek to construct. Here is the first, largely concerned with the symmetries of the fibration.

Assumption 7.1 Let $U$ be a connected open neighbourhood of $(0,0,0)$ in $\mathbin{\mathbb{C}}^{3}$ , which is invariant under the symmetries in parts (ii) and (v)–(viii) below. We aim to construct a special Lagrangian fibration $f:U\rightarrow\mathbin{\mathbb{R}}^{3}$ , with fibres $N_{a,b,c}=f^{-1}(a,b,c)$ , with the following properties:

(i)

$f$ is continuous, and smooth except on the real hypersurface $|z_{1}|=|z_{2}|$ .
(ii)

$f({\rm e}^{i\theta}z_{1},{\rm e}^{-i\theta}z_{2},z_{3})=f(z_{1},z_{2},z_{3})$ for all $(z_{1},z_{2},z_{3})\in U$ and $\theta\in\mathbin{\mathbb{R}}$ . Equivalently, every fibre $N_{a,b,c}$ is invariant under the $\mathbin{\rm U}(1)$ -action given by

${\rm e}^{i\theta}:(z_{1},z_{2},z_{3})\mapsto({\rm e}^{i\theta}z_{1},{\rm e}^{-i\theta}z_{2},z_{3}).$ (52)
(iii)

If $f(z_{1},z_{2},z_{3})=(a,b,c)$ then $a=|z_{1}|^{2}-|z_{2}|^{2}$ .
(iv)

The set of singular points of singular fibres of $f$ is $\bigl\{(0,0,z_{3})\in U\bigr\}$ . In particular, $N_{a,b,c}$ is nonsingular if $a\neq 0$ .
(v)

If $f(z_{1},z_{2},z_{3})=(a,b,c)$ then $f(z_{1},z_{2},z_{3}+it)=(a,b,c+t)$ for all $t\in\mathbin{\mathbb{R}}$ . This means that $N_{a,b,c+t}$ is the translation of $N_{a,b,c}$ by $(0,0,it)$ , and that

$N_{a,b,c}=\bigl\{(z_{1},z_{2},z_{3}+ic):(z_{1},z_{2},z_{3})\in N_{a,b,0}\bigr\}.$ (53)
(vi)

If $f(z_{1},z_{2},z_{3})=(a,b,c)$ then $f(z_{2},z_{1},z_{3})=(-a,b,c)$ .
(vii)

If $f(z_{1},z_{2},z_{3})=(a,b,c)$ then $f(\bar{z}_{1},\bar{z}_{2},\bar{z}_{3})=(a,b,-c)$ .
(viii)

If $f(z_{1},z_{2},z_{3})=(a,b,c)$ then $f(z_{1},z_{2},-z_{3})=(a,b,-c)$ .

Really we would like the domain $U$ of $f$ to be all of $\mathbin{\mathbb{C}}^{3}$ , and perhaps also to impose some asymptotic conditions on $f$ at infinity. But this would make our assumptions unnecessarily strong, and the author is not sure what asymptotic conditions would be appropriate. So instead we just suppose that $f$ is defined near $(0,0,0)$ . We will not worry very much about the issues raised by $f$ not being defined on all of $\mathbin{\mathbb{C}}^{3}$ , as they are primarily notational.

To understand where this list of properties has come from, note that the fibrations of Corollary 4.2 and Theorems 5.2 and 5.4 satisfy all of Assumption 7.1, except part (viii). We are aiming for a fibration that at a generic singular point is modelled one of the fibrations of Theorems 5.2 and 5.4, but will also have features in common with that of Corollary 4.2.

Therefore we simplify things by assuming that nearly all the symmetries these three fibrations have in common are also symmetries of the fibration we are aiming to construct. Here is our second assumption, drawing on the ideas of §6.

Assumption 7.2 Suppose that each fibre $N_{a,b,c}$ of $f$ may be written

\begin{split}N_{a,b,c}=\Bigl\{(&z_{1},z_{2},z_{3})\in U:\mathop{\rm Re}(z_{1}z_{2})=u_{a,b}\bigl(\mathop{\rm Re}(z_{3}),\mathop{\rm Im}(z_{1}z_{2})\bigr),\\ &\mathop{\rm Im}(z_{3})=v_{a,b}\bigl(\mathop{\rm Re}(z_{3}),\mathop{\rm Im}(z_{1}z_{2})\bigr)+c,\;\>|z_{1}|^{2}-|z_{2}|^{2}=a\Bigr\},\end{split}

(54)

where $u_{a,b},v_{a,b}:V_{a,b}\rightarrow\mathbin{\mathbb{R}}$ are 2-parameter families of functions and

V_{a,b}=\bigl\{\bigl(\mathop{\rm Re}(z_{3}),\mathop{\rm Im}(z_{1}z_{2})\bigr):(z_{1},z_{2},z_{3})\in N_{a,b,0}\bigr\}

(55)

is an open set in $\mathbin{\mathbb{R}}^{2}$ . Suppose also that the $u_{a,b},v_{a,b}$ satisfy:

(i)

$u_{0,b},v_{0,b}$ are smooth except at points $(x,0)$ in $V_{0,b}$ with $u_{0,b}(x,0)=0$ , and

\frac{\partial u_{0,b}}{\partial x}=-2\bigl(u_{0,b}^{2}+y^{2}\bigr)^{1/2}\frac{\partial v_{0,b}}{\partial y}\quad\text{and}\quad\frac{\partial u_{0,b}}{\partial y}=\frac{\partial v_{0,b}}{\partial x}

(56)

hold except at these points.

(ii)

When $a\neq 0$ , $u_{a,b}$ and $v_{a,b}$ are smooth on $V_{a,b}$ and satisfy

\frac{\partial u_{a,b}}{\partial x}=-\bigl(4u_{a,b}^{2}+4y^{2}+a^{2}\bigr)^{1/2}\frac{\partial v_{a,b}}{\partial y}\quad\text{and}\quad\frac{\partial u_{a,b}}{\partial y}=\frac{\partial v_{a,b}}{\partial x}.

(57)

(iii)

$u_{a,b}$ and $v_{a,b}$ depend continuously on $a,b$ , and smoothly wherever $a\neq 0$ .
(iv)

$u_{a,b}\equiv u_{-a,b}$ and $v_{a,b}\equiv v_{-a,b}$ for all $a,b$ .
(v)

$u_{a,b}(x,-y)=u_{a,b}(x,y)$ and $v_{a,b}(x,-y)=-v_{a,b}(x,y)$ for all $a,b,x,y$ .
(vi)

$u_{a,b}(-x,y)=u_{a,b}(x,y)$ and $v_{a,b}(-x,y)=-v_{a,b}(x,y)$ for all $a,b,x,y$ .

Here equation (54) essentially says that the fibres of $f$ may be written in the form (30). The explicit dependence on $c$ follows from part (v) of Assumption 7.1. Parts (i) and (ii) come from Proposition 6.1, and parts (iii)–(vi) from parts (i) and (vi)–(viii) of Assumption 7.1 respectively. Thus, the only thing Assumption 7.1 adds to Assumption 7.1 is that the fibres of $f$ may be written in the form (30).

Next we impose some conditions of a general topological nature which specify the ‘shape’ of the fibration we want, in particular the location and nature of its singularities.

Assumption 7.3 In the situation above, the functions $u_{a,b}$ and $v_{a,b}$ satisfy

(i)

For all $a,b$ , the function $u_{a,b}(x,0)$ is strictly increasing for $x<0$ and strictly decreasing for $x>0$ , with a maximum at 0.
(ii)

For all $a,b,b^{\prime},x,y$ with $b<b^{\prime}$ we have $u_{a,b}(x,y)<u_{a,b^{\prime}}(x,y)$ .
(iii)

Let $b>0$ , and write $b=\beta^{2}$ for $\beta>0$ . Then $u_{a,b}(\beta,0)=u_{a,b}(-\beta,0)=0$ for all $a$ . The solution $u_{0,b},v_{0,b}$ of (32) has isolated singularities of order 1 at $(\pm\beta,0)$ , in the sense of Definition 6.4.

Near $(-\beta,0)$ for small $a$ , the functions $u_{a,b},v_{a,b}$ are approximately equal to the functions $u,v$ constructed from $N_{a,-\beta}$ in Proposition 6.7.

Near $(\beta,0)$ for small $a$ , the functions $u_{a,b},v_{a,b}$ are approximately equal to the functions $u,v$ constructed from $N^{\prime}_{a,\beta}$ in Proposition 6.8.
(iv)

For all $a$ we have $u_{a,0}(0,0)=0$ , and the solution $u_{0,0},v_{0,0}$ of (32) has an isolated singularity of order 2 at $(0,0)$ , in the sense of Definition 6.4.
(v)

For all $a,x\in\mathbin{\mathbb{R}}$ and $b<0$ , we have $u_{a,b}(x,0)<0$ .

We will see in §7.2 that $u_{a,b}$ and $v_{a,b}$ actually depend only on the values of $u_{a,b}$ on the $x$ -axis. In Figure 1 we sketch the functions $u_{0,b}(x,0)$ for several values of $b$ , on the same graph, to display the general features we expect of these functions. The curves are smooth except where they intersect the $x$ -axis. The condition that $u_{0,b}(x,0)<u_{0,b^{\prime}}(x,0)$ when $b<b^{\prime}$ corresponds to the fact that the curves do not intersect, and move up the graph as $b$ increases.

Figure 1: approximate curves

u_{0,b}(x,0)

for different

b

For $\beta>0$ and $b=\beta^{2}$ , the curve $u_{0,b}(x,0)$ intercepts the $x$ -axis at $(\pm\beta,0)$ . By part (d) of Propositions 6.7 and 6.8 and part (iii) of Assumption 7.1, near $(-\beta,0)$ we have $u_{0,b}(x,0)\approx(x+\beta)|x+\beta|$ , and near $(\beta,0)$ we have $u_{0,b}(x,0)\approx-(x-\beta)|x-\beta|$ . Thus $u_{0,b}(x,0)$ is differentiable at $(\pm\beta,0)$ with gradient zero.

For comparison, in Figure 2 we sketch the functions $u_{a,b}(x,0)$ for some small fixed $a\neq 0$ , and the same values of $b$ . The general shapes of the curves are the same, as are the intercepts with the $x$ -axis. But the curves are all smooth, and using part (d) of Propositions 6.7 and 6.8 and part (iii) of Assumption 7.1, we see that

	$\displaystyle u_{a,b}(x,0)$	$\displaystyle\approx(x+\beta)\bigl((x+\beta)^{2}+\|a\|\bigr)^{1/2}$		near $x=-\beta$ , and
	$\displaystyle u_{a,b}(x,0)$	$\displaystyle\approx-(x-\beta)\bigl((x-\beta)^{2}+\|a\|\bigr)^{1/2}$		near $x=\beta$ ,

so that the gradient at $(-\beta,0)$ is approximately $|a|^{1/2}$ , and at $(\beta,0)$ approximately $-|a|^{1/2}$ . However, this approximation breaks down near $\beta=0$ .

Figure 2: curves

u_{a,b}(x,0)

for fixed small

a\neq 0

and different

b

Recall that our goal is to model a fibration in which generic singular fibres in codimension one have two singularities, each locally a $T^{2}$ -cone, but in codimension two these two cone points come together and fuse to form a new kind of singularity.

Assumption 7.1 implies that when $b=\beta^{2}$ for $\beta>0$ , the fibre $N_{0,b,c}$ will have two singular points at $(0,0,\pm\beta+ic)$ . Near $(0,0,-\beta+ic)$ it is locally modelled on the special Lagrangian $T^{2}$ -cone $N_{0,-\beta+ic}$ of Definition 5, and near $(0,0,\beta+ic)$ it is locally modelled on the SL $T^{2}$ -cone $N^{\prime}_{0,\beta+ic}$ of Definition 5.

When $b=0$ , the fibre $N_{0,0,c}$ has one singular point at $(0,0,ic)$ . It results from an isolated singular point of order 2 in $u_{0,0},v_{0,0}$ , so as in §6.4 we expect the tangent cone at $(0,0,ic)$ to be the union of two special Lagrangian 3-planes $\Pi_{\pm}$ intersecting in a real line. We cannot describe the singularity of $N_{0,0,c}$ much more explicitly without proving Conjecture 6.14.

When $b<0$ , the fibre $N_{0,b,c}$ is nonsingular. Thus the picture is that as $b$ decreases from positive to negative, two $T^{2}$ -cone singular points in $N_{0,b,c}$ come together, fuse to form a new kind of singularity, and then vanish. We can think of the two singular points in $N_{0,b,c}$ for $b>0$ as having opposite sign, so that when they come together they cancel out.

We can now formulate our main conjecture.

Conjecture 7.4

There exists an open neighbourhood $U$ of $(0,0,0)$ in $\mathbin{\mathbb{C}}^{3}$ and a special Lagrangian fibration $f:U\rightarrow\mathbin{\mathbb{R}}^{3}$ satisfying Assumptions 7.1–7.1.