2.7 Boundary condition of the ODE [022H]

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2.7 Boundary condition of the ODE

The ODE (7) is posed on $0<t<\infty$ , and we now discuss the boundary conditions to put at $t=0$ and $t=\infty$ . The infinity boundary can be reduced to the $t=0$ case by the symmetry of the ODE, which geometrically comes from the symmetry between $x_{1},x_{2}$ (up to elementary scaling). Geometrically one would like to find a partial completion of the generalized Calabi ansatz, near the infinity of $\bar{X}\setminus D_{1}\cup D_{2}$ . The generalized Calabi ansatz works in the neighbourhood of $D_{1}\cap D_{2}$ , and one would like to understand what happens near $D_{1}\setminus D_{2}$ and $D_{2}\setminus D_{1}$ .

The simplest boundary conditions to imagine is for the ODE to remain analytic at $t=0$ . In this case, one would specify $v(0)=v_{0}>0$ and $v^{\prime}(0)>-\frac{n+2}{n}v_{0}$ , and the ODE is non-singular in a neighbourhood of $t=0$ , so that $v$ has a Taylor expansion at $t=0$ . Geometrically, the size of $T^{2}$ remains of order $O(x_{1}^{\frac{2-n}{2n}})$ and the effective Kähler class of $Y$ remains of order $O(x_{1}^{2/n})$ as $t\to 0$ . There seems to be no known metric near $D_{1}\setminus D_{2}$ that one may attempt to glue to the generalized Calabi ansatz. Intuitively, one cannot suddenly ‘switch off’ the effective Kähler class at $t=0$ .

We are thus led to look for boundary conditions such that $\frac{n+2}{n}v+(1-t)v^{\prime}\to 0$ as $t\to 0$ , which intuitively means the effective Kähler class ‘switches off gradually’. The following type of boundary conditions is perhaps best motivated by fixing $v_{0}>0$ , and trying to decrease $v^{\prime}(0)$ until $v^{\prime}(0)+\frac{n+2}{n}v_{0}$ tends to zero. As $t\to 0$ , we require $v^{\prime}$ has some subleading power law behaviour

\begin{cases}&v=v_{0}+O(t),\\ &v^{\prime}=-\frac{n+2}{n}v_{0}+at^{\beta}+O(t),\\ &v^{\prime\prime}=a\beta t^{\beta-1}+O(1).\end{cases}

(9)

Here $0<\beta<1$ and $a>0$ are some constants to be determined. Thus

vv^{\prime\prime}-\frac{2}{n+2}v^{\prime 2}=av_{0}\beta t^{\beta-1}+O(1),

\frac{n+2}{n}v+(1-t)v^{\prime}=at^{\beta}+O(t),

and the ODE predicts

\beta-1+(n-2)\beta=0,\quad\beta=\frac{1}{n-1},

and

\frac{av_{0}}{n-1}a^{n-2}=\text{const},\quad a\propto v_{0}^{-1/(n-1)}.

Geometric meaning of the boundary condition

We can translate the boundary condition near $t=0$ into the asymptotic behaviour of the Calabi-Yau metric for $1\ll x_{2}\ll x_{1}$ . Using the second derivative computation (8), we get

\begin{cases}\frac{\partial^{2}u}{\partial x_{1}^{2}}\sim\frac{2(n+2)}{n^{2}}v_{0}x_{1}^{\frac{2}{n}-1},\\ \frac{\partial^{2}u}{\partial x_{1}\partial x_{2}}\sim-\frac{d_{1}}{d_{2}}\frac{2(n+2)}{n^{2}}v_{0}x_{1}^{\frac{2}{n}-1}\\ \frac{\partial^{2}u}{\partial x_{2}^{2}}\sim(\frac{d_{1}}{d_{2}})^{2}\frac{a}{n-1}t^{-\frac{n-2}{n-1}}x_{1}^{\frac{2}{n}-1}.\end{cases}

Recall that in the generalized Calabi-Yau ansatz, this Hessian matrix controls the base metric and the torus fibres. In the $x_{1}$ -direction of the base (i.e. the direction transverse to $D_{1}$ at infinity), and the $\log\xi_{1}$ circle direction, the behaviour is similar to what happens in the generic region where $t$ is of order one. However, in the $x_{2}$ direction, the base metric has a different scaling law:

\frac{\partial^{2}u}{\partial x_{2}^{2}}\sim(\frac{d_{1}}{d_{2}})^{2}\frac{a}{n-1}t^{-\frac{n-2}{n-1}}x_{1}^{\frac{2}{n}-1}=O(x_{1}^{\frac{2}{n}-\frac{1}{n-1}}x_{2}^{-\frac{n-2}{n-1}}).

The distance to $x_{2}\sim O(1)$ is $O(x_{1}^{\frac{1}{n}-\frac{1}{2(n-1)}}x_{2}^{\frac{n}{2(n-1)}})$ . The $\log\xi_{2}$ circle direction now has diameter of order

O(x_{2}^{-\frac{n-2}{2(n-1)}}x_{1}^{\frac{1}{n}-\frac{1}{2(n-1)}}),

which is much larger compared to the $\log\xi_{1}$ circle. In other words, the metric exhibits inhomogeneous collapsing phenomenon.

The potential is roughly

\begin{split}&u=x_{1}^{\frac{n+2}{n}}v(t)\approx x_{1}^{\frac{n+2}{n}}(v_{0}-\frac{n+2}{n}v_{0}t+\frac{n-1}{n}at^{\frac{n}{n-1}})\\ &\approx v_{0}(x_{1}-\frac{d_{1}}{d_{2}}x_{2})^{\frac{n+2}{n}}+\frac{n-1}{n}ax_{1}^{\frac{n+2}{n}}(\frac{d_{1}}{d_{2}}\frac{x_{2}}{x_{1}})^{\frac{n}{n-1}}.\end{split}

Here $x_{2}\ll x_{1}$ . For fixed values of $x_{1}-\frac{d_{1}}{d_{2}}x_{2}$ , the second term dominates the metric contribution on the slices. Up to scaling factors by powers of $x_{1}$ , the key dependence on $x_{2}$ is $dd^{c}x_{2}^{n/(n-1)}$ , which is the Calabi ansatz on an open subset of the $(n-1)$ -dimensional total space of the line bundle $\mathcal{O}(D_{2})\to D_{1}\cap D_{2}$ (cf. section 2.2). When $\xi_{1}$ is allowed to vary, the scales of the $S^{1}$ , the Calabi ansatz, and the base metric, all depend on $x_{1}$ in some power law fashion.

The significance to the compactification question, is that $\mathcal{O}(D_{2})\to D_{1}\cap D_{2}$ describes the infinity of the non-compact Calabi-Yau manifold $D_{1}\setminus D_{2}$ , and the Calabi ansatz is the asymptote of the Tian-Yau metric on $D_{1}\setminus D_{2}$ . We can thus glue the Calabi ansatz to the Tian-Yau metric, in a parametrized fashion over the $x_{1}$ variable, in order to achieve the partial completion of the generalized Calabi ansatz. After this, we would have an approximate Calabi-Yau metric outside a compact set in $X=\bar{X}\setminus D_{1}\cup D_{2}$ , from which one can hope to use a non-compact version of Yau’s proof to the Calabi conjecture to obtain an actual Calabi-Yau metric on $X$ .

Remark 2.7.

An appealing feature is that the boundary behaviour of the $m=2$ generalized Calabi ansatz is essentially the ordinary Calabi ansatz. We think this feature may generalize to larger $m$ , so that the boundaries have an inductive stratification structure.