4.2. Estimates on complex structures and metrics [03FJ]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

4.2. Estimates on complex structures and metrics

Let $J_{Z_{a}}$ and $J_{W_{a}}$ denote the complex structure operators on the tangent spaces to $Z_{a}$ and $W_{a}$ respectively. We would like to say that the embedding $\psi$ is holomorphic up to a small order terms.

We have already mentioned that $\psi$ is precisely holomorphic over the charts $V_{w}^{\beta^{\vee}}$ . To measure the discrepancy at $x\in U_{v}^{\beta}$ we will fix some (Euclidean) norm on $\mathbb{R}^{d-1}$ (they are all equivalent) to induce a norm on the tangent space $\mathbb{R}^{d-1}\oplus\mathbb{R}^{d-1}\cong T_{x}{W_{a}}$ . Let $C_{\lambda,\nu}(\beta)$ be the (uniform on $U_{v}^{\beta}$ ) bound for the $(\lambda,\nu)$ bi-PIKAS metric $g_{\lambda,\nu}$ written in the affine coordinates in $U_{v}^{\beta}$ .

Lemma 4.3.

As $\beta\to\infty,\beta^{\vee}\to 0$ , the linear map

d\psi\circ J_{Z_{a}}\circ(d\psi)^{-1}-J_{W_{a}}:T_{x}{W_{a}}\to T_{x}{W_{a}}

is of order $C_{\lambda-c,\nu}(\beta)\cdot O(\frac{1}{\beta^{\vee}}e^{-\beta})$ .

Proof.

First we apply estimates similar to those in Lemma 4.2 to the differential $d\psi$ , which we think of as an element in $(\mathbb{R}^{d}\oplus\mathbb{R}^{d})\otimes((\mathbb{R}^{d})^{*}\oplus(\mathbb{R}^{d})^{*})$ .

d\psi=\left(\begin{array}[]{cc}\mathbbm{1}-\mathfrak{X}\otimes v-(\langle v,\log|z|\rangle+\lambda(v))\nabla\mathfrak{X}&0\\ -(\langle v,\operatorname{Arg}(z)\rangle+\theta_{v})\nabla\mathfrak{X}&\mathbbm{1}-\mathfrak{X}\otimes v\end{array}\right)\left(\begin{array}[]{c}d\log|z|\\ d\operatorname{Arg}(z)\end{array}\right)

Note that the complex structures $J_{Z_{a}}$ and $J_{W_{a}}$ would match exactly via $d\psi$ if there were no $\nabla\mathfrak{X}$ terms (this is what happens in the charts $V_{w}^{\beta^{\vee}}$ where $\mathfrak{X}$ is constant).

According to (2) of Lemma 3.2 $\langle v,\mathfrak{X}\rangle=1$ in $U_{v}^{\beta}$ . Hence the projection operator

\left(\begin{array}[]{cc}\mathbbm{1}-\mathfrak{X}\otimes v&0\\ 0&\mathbbm{1}-\mathfrak{X}\otimes v\end{array}\right)

has a norm of order 1 when restricted to $Z_{a}$ , and the desired bound on $d\psi\circ J_{Z_{a}}\circ(d\psi)^{-1}-J_{W_{a}}$ will follow from estimating the $\nabla\mathfrak{X}$ terms.

But according to the Lemma 4.2 we have the uniform bounds:

|\langle v,\log|z|\rangle+\lambda(v)|\leq Ce^{-\beta},\ |\langle v,\operatorname{Arg}(z)\rangle+\theta_{v}|\leq Ce^{-\beta}.

On the other hand, by (3) of Lemma 3.2 the gradient $\nabla\mathfrak{X}(\log|z|)$ is bounded by $O(\frac{1}{\beta^{\vee}})\cdot\left|g_{\lambda-c,\nu}(\log|z|)\right|$ . ∎

The Kähler form on $Z_{a}^{\mathrm{sm}}$ is defined as the restriction of the Kähler form $\omega$ on $X_{T}$ which in $(\mathbb{C}\backslash\{0\})^{d}$ is given by:

\omega=\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\Phi^{\mathrm{sm}}_{\lambda+\gamma,\nu,h}\left(\log|z|\right)+\epsilon\omega_{0},

where $\omega_{0}$ is a fixed (e.g., the Fubini-Study) Kähler form. We will compare the metric induced by $\omega$ with the (degenerate) scalar product on $W_{a}$ induced by the $(\lambda,\nu)$ -bi-PIKAS.

To make these estimates we will need to introduce some bounds (in a Euclidean metric in $\mathbb{R}^{d}$ ) all of which follow essentially from the definition of the bi-PIKAS family:

\begin{split}&\left|g_{ij}(x)\right|<C_{0}(\beta),\text{ from Lemma~\ref{lemma:complex} above, uniformly for }x\in U_{v}^{\beta},\\ &\left|\operatorname{Hess}\Phi_{\lambda+\gamma}(x)\right|<C_{1}(\gamma),\text{ uniformly for }x\in\Delta^{\vee}_{\lambda+\gamma/2},\\ &\left|g_{\lambda+\gamma,\nu}(q)-g_{\lambda,\nu}(q)\right|<C_{2}(\gamma),\text{ uniformly for }q\in\partial\mathcal{U},\\ &\left|\operatorname{Hess}\Phi^{\mathrm{sm}}_{\lambda+\gamma,h}(x)-\operatorname{Hess}\Phi_{\lambda+\gamma}(x)\right|<C_{3}(h),\text{ uniformly for }x\in\Delta^{\vee}_{\lambda+\gamma/2},\\ &\left|\operatorname{Hess}\Phi_{\lambda+\gamma}|_{\mathbb{R}^{d}_{v}}(q+t\mathfrak{X}(x))-\operatorname{Hess}\Phi_{\lambda}|_{\mathbb{R}^{d}_{v}}(q)\right|<C_{4}(\gamma),\text{ uniformly for }\\ &\hskip 72.26999pt-\gamma<t<\gamma,q\in U_{v}^{\beta}\subset\partial\Delta^{\vee}_{\lambda},\\ &\left|\operatorname{Hess}\Phi_{\lambda}(x)-\operatorname{Hess}K_{w}(q)\right|<C_{5}(\beta,c),\text{ uniformly for }\\ &\hskip 72.26999ptq\in N^{\beta}(\partial\mathcal{U})\cap V_{w}^{\beta^{\vee}},x\in\mathcal{F}_{q}:=q+\mathbb{R}_{\geq-c}\cdot w,\end{split}

where $C_{0}(\beta)\to\infty,\ C_{1}(\gamma)\to\infty,\ C_{2}(\gamma)\to 0,\ C_{3}(h)\to 0,\ C_{4}(t)\to 0,\ C_{5}(\beta,c)\to 0$ as (all of) the corresponding parameters go to 0. The last inequality follows from $\Phi_{\lambda}\in C^{2}(\partial\mathcal{U}\backslash D)$ , where the local potential $K_{w}(x)$ is pulled back from the quotient.

Lemma 4.4.

Under the embedding $\psi:Z_{a}^{\mathrm{sm}}\to W_{a}$ the scalar products agree up to terms of order $C_{2}(\gamma)+C_{3}(h)+C_{1}(\gamma)\frac{C_{0}(\beta)}{\beta^{\vee}}e^{-\beta}+C_{4}(\gamma)+C_{5}(\beta,c)+O(\epsilon).$

Proof.

Let $x\in X(U_{v}^{\beta})\cap Z_{a}$ . Assuming $e^{-\beta}<\gamma/2$ we have

\left|\operatorname{Hess}\Phi^{\mathrm{sm}}_{\lambda+\gamma,h}(x)-\operatorname{Hess}\Phi_{\lambda+\gamma}(x)\right|<C_{3}(h).

The difference between $\operatorname{Hess}\Phi_{\lambda+\gamma}|_{T_{x}Z_{a}}$ and $\operatorname{Hess}\Phi_{\lambda+\gamma}|_{T_{\psi(x)}W_{a}}$ consists of two terms. The first term $C_{4}(\gamma)$ appears from comparing $\operatorname{Hess}\Phi_{\lambda+\gamma}|_{\mathbb{R}^{d}_{v}}$ at $q+t\mathfrak{X}(x)$ with $\operatorname{Hess}\Phi_{\lambda}|_{\mathbb{R}^{d}_{v}}$ at $q$ . The other term $C_{1}(\gamma)\frac{C_{0}(\beta)}{\beta^{\vee}}e^{-\beta}$ reflects the error in the alignment of the tangent spaces via the map $d\psi$ in the proof of Lemma 4.3.

Now let $x\in X(V_{w}^{\beta^{\vee}})$ . We will just need to check points in $\mathcal{F}_{q}$ for $q\in V_{w}^{\beta^{\vee}}\backslash\bigcup U_{v}^{\beta}=N^{\beta}(\partial\mathcal{U})\cap V_{w}^{\beta^{\vee}}$ . Note that $\operatorname{Hess}\Phi_{\lambda}$ is bounded in $\mathcal{F}(V_{w}^{\beta^{\vee}})$ and is continuous at $\partial\mathcal{U}\backslash D$ (in particular, the Hessian vanishes into the $w$ -direction at $\partial\mathcal{U}\backslash D$ ). Hence the $C_{5}$ bound from above are also valid for the regularization $\operatorname{Hess}\Phi^{\mathrm{sm}}_{\lambda+\gamma,h}$ . Namely,

\left|\operatorname{Hess}\Phi^{\mathrm{sm}}_{\lambda+\gamma,h}(x)-g_{\lambda+\gamma,\nu}(q)\right|<C_{5}(\beta,c),

with possibly different function $C_{5}$ . The discrepancy between $g_{\lambda+\gamma,\nu}$ and $g_{\lambda,\nu}$ in $N^{\beta}(\partial\mathcal{U})\cap V_{w}^{\beta^{\vee}}$ is encoded in the $C_{2}(\gamma)$ term.

Finally, the term $\epsilon\omega_{0,\nu}$ in $\omega$ can be bounded by $O(\epsilon)$ . ∎