4.3. The Gromov-Hausdorff limits of one-parameter families [03FP]

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4.3. The Gromov-Hausdorff limits of one-parameter families

We will apply the results of the previous sections to the situation considered in [HZ02] to draw a consequence mostly related to the mirror symmetry conjecture. Let $\lambda_{0}$ be an integral vector in the interior of the secondary cone $\operatorname{SC}(S)$ . We consider an 1-parameter family of the hypersurfaces $Z_{s}$ defined as closures in $X_{T}$ of

Z_{s}^{\operatorname{af{}f}}:=\{z\in(\mathbb{C}\backslash\{0\})^{d}\ :\ \sum_{m\in\Delta_{\mathbb{Z}}\backslash\{0\}}a_{m}s^{\lambda_{0}(m)}z^{m}=1\}.

Choose an integral vector $\nu_{0}$ in the interior of $\operatorname{SC}(T)$ and consider $\Sigma$ with the metric space structure given by the bi-PIKAS $(\lambda_{0},\nu_{0})$ .

Also consider an one-parameter family of (non-compact) Kähler manifolds $W_{s}:=W(\log|a|+\log|s|\cdot\lambda_{0},\operatorname{Arg}(a)+\operatorname{Arg}(s)\cdot\lambda_{0})$ , whose metric and complex structure are induced from the $\left(\log|a|+\log|s|\cdot\lambda_{0},\frac{\nu_{0}}{\log|s|}\right)$ bi-PIKAS.

Theorem 4.5.

As $|s|\to\infty$ one can choose smooth portions of the hypersurfaces $Z_{s}^{\mathrm{sm}}\subset Z_{s}$ , the embeddings $\psi_{s}:Z_{s}^{\mathrm{sm}}\hookrightarrow W_{s}$ and a family of Kähler metrics on $Z_{s}$ in the class $\frac{\nu_{0}}{\log|s|}(1+o(1))$ such that the pairs $(Z_{s},Z_{s}\backslash Z_{s}^{\mathrm{sm}})$ converges to the pair $(\Sigma,D)$ in the Gromov-Hausdorff sense, and the maps $\psi_{s}$ identify (uniformly in $x\in Z_{s}^{\mathrm{sm}}$ ) the scalar products and the complex structures on the tangent spaces $T_{x}Z_{s}$ and $T_{\psi_{s}(x)}W_{s}$ up to terms of order $o(1)$ .

Proof.

We consider the bi-PIKAS family in a neighborhood of $(\lambda_{0},\nu_{0})$ and extend it by rescaling to (a neighborhood of) the ray $(\lambda,\nu)=\left(\log|a|+\log|s|\cdot\lambda_{0},\frac{\nu_{0}}{\log|s|}\right)$ in $\operatorname{SC}(S)\times\operatorname{SC}(T)$ .

The $C_{i}$ -estimates for the $(\lambda,\nu)$ bi-PIKAS considered in the bi-PIKAS metric (rather than in Euclidean) are equivalent to the corresponding estimates for the rescaled structure $((\log|s|)^{-1}\lambda,\log|s|\nu)$ made in the Euclidean metric as before. This is because the Euclidean metric on $\partial\Delta_{\lambda_{0}}$ is equivalent to the bi-PIKAS metric on $\Sigma$ . But to pass from $(\lambda,\nu)$ to $((\log|s|)^{-1}\lambda,\log|s|\nu)$ we will need to rescale all the parameters as well:

	$\displaystyle\lambda\sim\lambda_{0}\log\|s\|,\ \beta\sim\beta_{0}\log\|s\|,\ \gamma\sim\gamma_{0}\log\|s\|,\ h\sim h_{0}\log\|s\|,\ c\sim c_{0}\log\|s\|,$
	$\displaystyle\nu\sim\frac{\nu_{0}}{\log\|s\|},\ \beta^{\vee}\sim\frac{\beta^{\vee}_{0}}{\log\|s\|},\ \operatorname{Hess}\Phi\sim\frac{\operatorname{Hess}\Phi_{0}}{(\log\|s\|)^{2}}.$

Or equivalently, we could apply the log map with the base $|s|$ as in [HZ02].

We saw in the proof of Lemma 4.4 that $\operatorname{Hess}\Phi^{\mathrm{sm}}_{\lambda_{0}+\gamma_{0}}$ is degenerate along $\mathcal{F}$ outside $\Delta^{\vee}_{\lambda_{0}+\gamma_{0}+h_{0}}$ . This argument extended to the entire toric variety shows that up to terms of order $O(\epsilon)$ , the set $X_{T}\backslash\log^{-1}(\Delta^{\vee}_{\lambda_{0}+\gamma_{0}+h_{0}})$ (which contains $Z_{s}$ ) has distance from $\partial\Delta^{\vee}_{\lambda}$ bounded by the diameters of the torus fibers $\mathbb{T}$ . The size of the tori $\mathbb{T}$ is determined by the norm of $\operatorname{Hess}\Phi^{\mathrm{sm}}_{\lambda+\gamma}$ at the corresponding point, which is bounded by $\frac{C_{1}(\gamma_{0})}{(\log|s|)^{2}}$ .

The rest of the proof consists of careful picks for asymptotics of the rescaled parameters $\beta_{0},\beta^{\vee}_{0},h,\gamma_{0},c_{0},\epsilon$ to ensure that the following expressions

	$\displaystyle\beta_{0},\quad\beta^{\vee}_{0},\quad\gamma_{0},\quad c_{0},\quad h_{0},$
	$\displaystyle C_{1}(\gamma_{0})(\log\|s\|)^{-2},$
	$\displaystyle\log\|s\|\frac{1}{\beta^{\vee}_{0}}e^{-\beta_{0}\log\|s\|}\cdot C_{0}(\beta_{0}),$
	$\displaystyle\log\|s\|\frac{C_{0}(\beta_{0})C_{1}(\gamma_{0})}{\beta^{\vee}_{0}}e^{-\beta_{0}\log\|s\|}+C_{2}(\gamma_{0})+C_{3}(h_{0})$
	$\displaystyle\hskip 144.54pt+C_{4}(\gamma_{0})+C_{5}(\beta_{0},c_{0})+O(\epsilon)(\log\|s\|)^{2}$

go to 0 as $\log|s|\to\infty$ , where the first two lines take care of the Hausdorff convergence, and the last two give matching of the complex structure and the metric under the embedding $\psi:Z^{\mathrm{sm}}_{s}\to W_{s}$

For instance, we can choose

\beta^{\vee}_{0}\sim\frac{1}{\log|s|},\quad\gamma_{0}\sim C_{1}^{-1}(\log|s|),\quad c_{0}\sim\frac{1}{\log|s|},\quad h_{0}\sim\frac{1}{\log|s|}.

And $\beta_{0}(\log|s|)$ has satisfy $C_{1}(\frac{1}{\log|s|}e^{-\beta_{0}\log|s|})<\log|s|$ (i.e. $e^{-\beta}<\gamma$ , which is needed for the proof of Lemma 4.4) and

(\log|s|)^{3}C_{0}(\beta_{0})e^{-\beta_{0}\log|s|}\to 0,

which is possible due to the fast decreasing factor of $e^{-\beta_{0}\log|s|}$ when $\beta_{0}(\log|s|)$ is changing slowly.

Finally, notice that the bi-PIKAS of type $((\log|s|)^{-1}\cdot\lambda,\log|s|\cdot\nu)=(\lambda_{0}+(\log|s|)^{-1}\log|a|,\nu_{0})$ converges to the $(\lambda_{0},\nu_{0})$ -bi-PIKAS. ∎

Remark.

We can rephrase the above theorem in terms of the alternate definition of the torus bundles $W_{\frac{1}{\log|s|}}(\lambda_{0},\nu_{0})$ associated to the given Kähler affine structure on $\Sigma$ . Then the statement of the theorem will coincide with the Conjecture 2 of [KS01].

	$\displaystyle\beta_{0},\quad\beta^{\vee}_{0},\quad\gamma_{0},\quad c_{0},\quad h_{0},$
	$\displaystyle C_{1}(\gamma_{0})(\log\|s\|)^{-2},$
	$\displaystyle\log\|s\|\frac{1}{\beta^{\vee}_{0}}e^{-\beta_{0}\log\|s\|}\cdot C_{0}(\beta_{0}),$
	$\displaystyle\log\|s\|\frac{C_{0}(\beta_{0})C_{1}(\gamma_{0})}{\beta^{\vee}_{0}}e^{-\beta_{0}\log\|s\|}+C_{2}(\gamma_{0})+C_{3}(h_{0})$
	$\displaystyle\hskip 144.54pt+C_{4}(\gamma_{0})+C_{5}(\beta_{0},c_{0})+O(\epsilon)(\log\|s\|)^{2}$