8.1 A variation on the fibration of § 7.1 [03ME]

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8.1 A variation on the fibration of §7.1

Consider a fibration $f$ satisfying the following analogues of Assumptions 7.1–7.1.

Assumption 8.1 Let $\mathbin{\mathbb{Z}}$ act on $\mathbin{\mathbb{C}}^{3}$ by $(z_{1},z_{2},z_{3})\,{\mathrel{\mathop{\kern 0.0pt\mapsto}\limits^{n}}}\,(z_{1},z_{2},z_{3}+2\pi n)$ for $n\in\mathbin{\mathbb{Z}}$ . Define $U=\bigl\{(z_{1},z_{2},z_{3}\!+\!2\pi\mathbin{\mathbb{Z}})\in\mathbin{\mathbb{C}}^{3}/\mathbin{\mathbb{Z}}:\mathop{\rm Im}(z_{1}z_{2})\in[-\pi,\pi]\bigr\}$ . Consider a special Lagrangian fibration $f:U\rightarrow\mathbin{\mathbb{R}}^{3}$ , with fibres $N_{a,b,c}=f^{-1}(a,b,c)$ . Suppose that $f$ satisfies parts (i)–(viii) of Assumption 7.1.

Assumption 8.2 Suppose that each fibre $N_{a,b,c}$ of $f$ may be written

	$\displaystyle N_{a,b,c}=\Bigl\{($	$\displaystyle z_{1},z_{2},z_{3}+2\pi\mathbin{\mathbb{Z}})\in U:\mathop{\rm Re}(z_{1}z_{2})=u_{a,b}\bigl(\mathop{\rm Re}(z_{3})+2\pi\mathbin{\mathbb{Z}},\mathop{\rm Im}(z_{1}z_{2})\bigr),$
		$\displaystyle\mathop{\rm Im}(z_{3})=v_{a,b}\bigl(\mathop{\rm Re}(z_{3})+2\pi\mathbin{\mathbb{Z}},\mathop{\rm Im}(z_{1}z_{2})\bigr)+c,\;\>\|z_{1}\|^{2}-\|z_{2}\|^{2}=a\Bigr\},$

where $u_{a,b},v_{a,b}$ are functions $(\mathbin{\mathbb{R}}/2\pi\mathbin{\mathbb{Z}})\times[-\pi,\pi]\rightarrow\mathbin{\mathbb{R}}$ . Suppose also that $u_{a,b}$ and $v_{a,b}$ satisfy parts (i)–(vi) of Assumption 7.1.

Assumption 8.3 In the situation above, the functions $u_{a,b}$ and $v_{a,b}$ satisfy

(i)

For all $a,b$ , the function $u_{a,b}(x+2\pi\mathbin{\mathbb{Z}},0)$ is strictly increasing for $x$ in $(-\pi,0)$ and strictly decreasing for $x$ in $(0,\pi)$ , with a maximum at $2\pi\mathbin{\mathbb{Z}}$ and a minimum at $\pi+2\pi\mathbin{\mathbb{Z}}$ .
(ii)

For all $a,b,b^{\prime},x,y$ with $b<b^{\prime}$ we have $u_{a,b}(x+2\pi\mathbin{\mathbb{Z}},y)<u_{a,b^{\prime}}(x+2\pi\mathbin{\mathbb{Z}},y)$ .
(iii)

For all $a,b,x\in\mathbin{\mathbb{R}}$ we have $u_{a,b}(x+2\pi\mathbin{\mathbb{Z}},0)=0$ if and only if $b+\cos x=0$ .
(iv)

Let $b\in(-1,1)$ , and write $b=\cos\beta$ for $\beta\in(0,\pi)$ . Then the solution $u_{0,b},v_{0,b}$ of (32) has isolated singularities of order 1 at $(\pm\beta+2\pi\mathbin{\mathbb{Z}},0)$ , in the sense of Definition 6.4.

Near $(-\beta+2\pi\mathbin{\mathbb{Z}},0)$ 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 $(\beta+2\pi\mathbin{\mathbb{Z}},0)$ 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.
(v)

The solution $u_{0,-1},v_{0,-1}$ of (32) has an isolated singularity of order 2 at $(2\pi\mathbin{\mathbb{Z}},0)$ , and the solution $u_{0,1},v_{0,1}$ has an isolated singularity of order 2 at $(\pi+2\pi\mathbin{\mathbb{Z}},0)$ , in the sense of Definition 6.4.
(vi)

For all $a,b,x,y\in\mathbin{\mathbb{R}}$ we have $u_{a,-b}(x+2\pi\mathbin{\mathbb{Z}},y)=-u_{a,b}(x+\pi+2\pi\mathbin{\mathbb{Z}},y)$ and $v_{a,-b}(x+2\pi\mathbin{\mathbb{Z}},y)=-v_{a,b}(x+\pi+2\pi\mathbin{\mathbb{Z}},y)$ .

These assumptions are a kind of toy model, designed to illustrate some aspects of how the fibrations of §5 and §7 might fit together in a Calabi–Yau 3-fold, and to perform a topological calculation. We are not making the conjecture that a fibration actually exists satisfying these assumptions.

Figure 3: approximate curves

u_{0,b}(x,0)

for different

b

In Figure 3 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 basic idea is that $u_{0,b}(x,0)$ should look a bit like $b+\cos x$ , in that it has the same periodic behaviour, is zero at the same points, and is increasing and decreasing in the same regions. But at its zeros $u_{0,b}(x,0)$ has gradient zero, whereas $b+\cos x$ generally does not.

Let us describe the fibres $N_{a,b,c}$ of $f$ . From §6, $N_{a,b,c}$ is singular if and only if $a=0$ and $u_{0,b}(x,0)=0$ for some $x$ . But part (iii) of Assumption 8.1 shows that $u_{0,b}(x+2\pi\mathbin{\mathbb{Z}},0)=0$ if and only if $\cos x=-b$ . This has solutions when $b\in[-1,1]$ . Therefore $N_{a,b,c}$ is singular if and only if $a=0$ and $b\in[-1,1]$ .

It is easy to show that the nonsingular fibres of $f$ are all diffeomorphic to $T^{2}\times[-\pi,\pi]$ . The singular fibres $N_{0,b,c}$ for $b\in(-1,1)$ are diffeomorphic to $T^{2}\times[-\pi,\pi]$ with two homologous circles collapsed to two points, and the singular fibres $N_{0,\pm 1,c}$ are diffeomorphic to $T^{2}\times[-\pi,\pi]$ with one circle collapsed to a point.