4.5. Structure near the singular locus II [045J]

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4.5. Structure near the singular locus II

This Section interprets the metric structure transverse to the singular locus $S$ in terms of Taub-NUT metrics. First we emphasize that smooth topology of the $S^{1}$ -fibration is subtle.

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The Kähler ansatz is only defined a priori on the complement of $S$ , and there are no transparent choices of smooth local coordinates near $S$ to exhibit the smooth extension of both the complex structure and the metric.
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By the reasons stated in Remark 2.4, smooth topology of the singular $S^{1}$ -bundle is expected to be unstable under deformation.

Because of these difficulties, in the region $\{R\lesssim A^{-1/2}\}$ near the singular locus $S\cap M^{-}$ , we will be forced to work with tensor fields of low regularity.

We now define a model metric

(4.20)

\begin{cases}g_{\text{NUT}}=(A+\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}})(d\mu^{2}+|d\xi_{1}|^{2})+(A+\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}})^{-1}\vartheta_{\text{NUT}}^{2}+A|d\xi_{2}|^{2},\\ \omega_{\text{NUT}}=(A+\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}})(d\mu\wedge\vartheta_{\text{NUT}}+\frac{\sqrt{-1}}{2}d\xi_{1}\wedge d\bar{\xi}_{1})+\frac{A\sqrt{-1}}{2}d\xi_{2}\wedge d\bar{\xi}_{2},\\ \Omega_{\text{NUT}}=A^{1/2}(\vartheta_{\text{NUT}}-\sqrt{-1}(A+\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}})d\mu)\wedge d\xi_{1}\wedge d\xi_{2},\end{cases}

namely the product of the Taub-NUT metric with flat $\mathbb{R}^{2}$ . The weighted Hölder norm $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta}(g_{\text{NUT}})}$ for $S^{1}$ -invariant tensors $T$ on this model space is defined by

\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta}(g_{\text{NUT}})}=\sup_{p}(A^{1/2}\ell)^{-\delta}\{\sum_{j=0}^{k}|\ell^{j}\nabla^{j}T(p)|+\sup_{|p-p^{\prime}|_{a}\leq\ell}\frac{|\nabla^{k}T(p)-\nabla^{k}T(p^{\prime})|}{d_{g_{\text{NUT}}}(p,p^{\prime})^{\alpha}\ell^{-\alpha-k}}\}.

where $\ell\sim A^{1/2}\sqrt{\mu_{1}^{2}+|\xi_{1}|^{2}}+A^{-1/2}$ reflects the regularity scale of the model metric, and we compare $T(p)$ and $T(p^{\prime})$ using parallel transport along minimal geodesics. An estimate in this norm is thought as the Hölder version of $|T|=O((A^{1/2}\ell)^{\delta}).$

Proposition 4.18.

Via the local diffeomorphism $\Psi$ , on the chart $\{r\lesssim A^{1/4}\}$ ,

\begin{cases}|\nabla_{g_{a}}\{\Psi^{*}(w^{p\bar{q}}d\eta_{p}\otimes d\bar{\eta}_{q})-\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}d\xi_{1}\otimes d\bar{\xi}_{1}\}|_{g_{a}}\leq CA^{-1},\\ |\nabla_{g_{a}}\{\Psi^{*}v-\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}\}|_{g_{a}}\leq C.\end{cases}

Furthermore

\begin{cases}|\nabla^{2}_{g_{a}}\{\Psi^{*}(w^{p\bar{q}}d\eta_{p}\otimes d\bar{\eta}_{q})-\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}d\xi_{1}\otimes d\bar{\xi}_{1}\}|_{g_{\text{NUT}}}\leq CA^{-3/4}\ell^{-2},\\ |\nabla^{2}_{g_{a}}\{\Psi^{*}v-\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}\}|_{g_{\text{NUT}}}\leq CA^{1/4}\ell^{-2}.\end{cases}

Proof.

(Sketch) The method is the same as in Proposition 4.16, so we only mention the key points. The long distance contribution to the integral has improved decay, so there is no need for the log factor. The short distance contribution appeals to Lemma 4.15. In the first derivative estimates, notice the mean curvature vector vanishes because $S$ an algebraic curve.

For second derivative estimates, notice for $R\lesssim A^{-1/2}$ , the magnitudes of tensors are inhomogeneous:

|d\mu|_{g_{\text{NUT}}}\leq A^{-1/4}R^{1/2},\quad|d\xi_{1}|_{g_{\text{NUT}}}\leq A^{-1/4}R^{1/2},\quad|d\xi_{2}|_{g_{\text{NUT}}}\leq CA^{-1/2}.

These $R$ factors make the $g_{\text{NUT}}$ -magnitudes bounded even though the $g_{a}$ -magnitudes can be unbounded. Inside the tensor $\Psi^{*}(w^{p\bar{q}}d\eta_{p}\otimes d\bar{\eta}_{q})$ , the coeffient of $d\xi_{1}\otimes d\bar{\xi}_{2}$ and $d\xi_{2}\otimes d\bar{\xi}_{1}$ correspond to imposing $f(0)=0$ , and the coeffient of $d\xi_{2}\otimes d\bar{\xi}_{2}$ correspond to imposing $f(0)=0$ and $df(0)=0$ . ∎

Proposition 4.19.

(Transverse Taub-NUT metric) Fix $0<\alpha<1$ . Under suitable gauge choices for the $S^{1}$ -connection, the ansatz metric $g^{(1)}$ over the local chart $\{r\lesssim A^{1/4}\}$ is approximated by the model metric:

\begin{cases}|\Psi^{*}g^{(1)}-g_{\text{NUT}}|_{g_{\text{NUT}}}\leq CA^{-3/4}\max(\log(A^{-1/4}\varrho),1)+CA^{1/4}|\xi_{2}|,\\ |\nabla_{g_{\text{NUT}}}(\Psi^{*}g^{(1)}-g_{\text{NUT}})|_{C^{\alpha}_{-1}(r\lesssim A^{1/4})}\leq CA^{-1/4}\max(\log(A^{-1/4}\varrho),1).\end{cases}

and

\begin{cases}|\Psi^{*}\Omega^{(1)}-\Omega_{\text{NUT}}|_{g_{\text{NUT}}}\leq CA^{-3/4}\max(\log(A^{-1/4}\varrho),1)+CA^{1/4}|\xi_{2}|,\\ |\nabla_{\Omega_{\text{NUT}}}(\Psi^{*}g^{(1)}-\Omega_{\text{NUT}})|_{C^{\alpha}_{-1}(r\lesssim A^{1/4})}\leq CA^{-1/4}\max(\log(A^{-1/4}\varrho),1).\end{cases}

Proof.

Applying the asymptotes in Proposition 4.16 and 4.18, on the local chart $\{r\lesssim A^{1/4}\}$ , up to an error of order $O(A^{-3/4}\max(1,\log(A^{-1/4}\varrho)))$ , the ansatz metric admits asymptote

(4.21)

\begin{cases}\begin{split}g^{(1)}=&V_{(1)}d\mu^{2}+V_{(1)}^{-1}\vartheta^{2}+\text{Re}(W^{p\bar{q}}_{(1)}d\eta_{p}\otimes d\bar{\eta}_{q})\\ \sim&\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}(d\mu^{2}+|d\xi_{1}|^{2})+(A+\frac{1}{2\sqrt{\mu^{2}+|\xi|^{2}}})^{-1}\vartheta^{2}+g_{a},\end{split}\\ \omega^{(1)}\sim d\mu\wedge\vartheta+\frac{\sqrt{-1}}{2}(a_{p\bar{q}}d\eta_{p}\wedge d\bar{\eta}_{q}+\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}}d\xi_{1}\wedge d\bar{\xi}_{1}),\\ \Omega^{(1)}\sim A^{1/2}\{\vartheta-\sqrt{-1}(A+\frac{1}{2\sqrt{\mu^{2}+|\xi_{1}|^{2}}})d\mu)\}\wedge d\eta_{1}\wedge d\eta_{2}.\end{cases}

The main issue then is to compare the connection $\vartheta$ with $\vartheta_{\text{NUT}}$ . The curvature of the Taub-NUT metric $d\vartheta_{\text{NUT}}$ has the explicit formula

\sqrt{-1}\{\frac{-\mu}{4(\mu^{2}+|\xi_{1}|^{2})^{3/2}}d\xi_{1}\wedge d\bar{\xi}_{1}+\frac{-\bar{\xi}_{1}}{4(\mu^{2}+|\xi_{1}|^{2})^{3/2}}d\mu\wedge d\xi_{1}-\frac{-\xi_{1}}{4(\mu^{2}+|\xi_{1}|^{2})^{3/2}}d\mu\wedge d\bar{\xi}_{1}\}.

The curvature $d\vartheta$ is prescribed by formula (1.6), involving the first derivatives of $v$ and $w^{p\bar{q}}$ . Applying the asymptotic from Proposition 4.18,

|(d\vartheta-d\vartheta_{\text{NUT}})|_{g_{a}}\leq CA^{-1/2},\quad|\nabla_{g_{a}}(d\vartheta-d\vartheta_{\text{NUT}})|_{g_{\text{NUT}}}\leq CA^{-1/4}\ell^{-2}.

In particular $\left\lVert(d\vartheta-d\vartheta_{\text{NUT}})\right\rVert_{C^{1}_{-1}(g_{\text{NUT}})}\leq CA^{1/4}$ . After suitable gauge fixing, we can find a 1-form $\vartheta-\vartheta_{\text{NUT}}$ on the base $\{r\lesssim A^{1/4}\}$ , with norm estimate upstairs $\left\lVert\vartheta-\vartheta_{\text{NUT}}\right\rVert_{C^{1,\alpha}_{0}(g_{\text{NUT}})}\leq CA^{-1/4}$ for any fixed $0<\alpha<1$ . This specifies a gauge choice of $\vartheta$ .

Thus up to an admissible amount of error we can replace $\vartheta$ by $\vartheta_{\text{NUT}}$ in the asymptote (4.21). The deviation between RHS of (4.21) and $g_{\text{NUT}}$ is an elementary term $\Psi^{*}g_{a}-A(d\mu^{2}+|d\xi_{1}|^{2}+|d\xi_{2}|^{2})$ controlled by (4.19). ∎

Remark 4.5.

For $R\gtrsim A^{-1/2}$ , one can use $\Delta_{a}$ -harmonicity to obtain weighted $C^{k,\alpha}$ -estimates for any large $k$ . For $R\lesssim A^{-1/2}$ , we exhibited a collection of local charts corresponding to a choice of $P\in S$ , such that the Kähler ansatz has $C^{1,\alpha}$ -regularity. Thus if $u$ is a $C^{2,\alpha}$ function on a chart, then its $g^{(1)}$ -Laplacian is $C^{\alpha}$ . This regularity is sufficient for setting up weighted Hölder analysis.