Proof. See [Reference Zhang47, Lemma 3.1, Th. 3.6] for (1) and [Reference Zhang48, Lemma 2.1] for (2).

Proof. (1) Let $\Xi \in H^{0}(Y, K_{Y}^{\otimes 2})\setminus \{0\}$ and $\Xi ' \in H^{0}(Y', K_{Y'}^{\otimes 2})\setminus \{0\}$ . Then $\varphi ^{*}\Xi $ is a bicanonical from on $Y'\setminus (\mathrm {Sing}\,Y' \cup \varphi ^{-1}(\mathrm {Sing}\,Y))$ , and $\Xi '$ is nowhere vanishing. We get $\varphi ^{*}\Xi /\Xi ' \in {\mathcal O}(Y'\setminus (\mathrm {Sing}\,Y' \cup \varphi ^{-1}(\mathrm {Sing}\,Y))) = {\mathcal O}(Y'\setminus \varphi ^{-1}(\mathrm {Sing}\,Y)) = {\mathcal O}(Y\setminus \mathrm {Sing}\,Y) = {\mathcal O}(Y)={\mathbf C}$ , where the first and third equalities follow from the normality of $Y'$ and Y and the second equality follows from the Zariski Main Theorem. Hence $\varphi ^{*}\Xi = c\,\Xi '$ with some $c \in {\mathbf C}\setminus \{0\}$ , and $\varphi ^{*}$ is an isomorphism.

(2) Let $o \in \mathrm {Sing}\,Y'$ . Assume $\varphi (o) \in Y \setminus \mathrm {Sing}\,Y$ . There exist a neighbourhood U of $\varphi (o)$ and a nowhere vanishing canonical form $\eta \in H^{0}(U, K_{Y})$ . We can express $\Xi |_{U} = F\cdot \eta ^{\otimes 2}$ , $F \in {\mathcal O}^{*}(U)$ . Since $\varphi ^{*}\Xi $ and $\varphi ^{*}F$ are nowhere vanishing on $\varphi ^{-1}(U)$ , so is $\varphi ^{*}\eta ^{\otimes 2}$ . Hence $\varphi ^{*}\eta $ is nowhere vanishing. Since any singular point of $Y'$ is non-Gorenstein, we get a contradiction. Thus $\varphi (o) \in \mathrm {Sing}\,Y$ .

(3) Since $\varphi ^{*}\Xi $ is nowhere vanishing on $Y'\setminus \varphi ^{-1}(\mathrm {Sing}\,Y)$ , $\varphi $ has no critical points on $Y'\setminus \varphi ^{-1}(\mathrm {Sing}\,Y)$ . Since the restriction of $\varphi $ to $Y'\setminus \varphi ^{-1}(\mathrm {Sing}\,Y)$ is a closed map, $\varphi \colon Y'\setminus \varphi ^{-1}(\mathrm {Sing}\,Y) \to Y \setminus \mathrm {Sing}\,Y$ is an étale covering of degree one: that is, an isomorphism. $\varphi $ induces a holomorphic map $f \colon X' \setminus (p')^{-1}\varphi ^{-1}(\mathrm {Sing}\,Y) \to X\setminus p^{-1}(\mathrm {Sing}\,Y)$ such that $p\circ f = \varphi \circ p'$ . Since $p^{-1}(y)$ consists of a unique point for any $y\in \mathrm {Sing}\,Y$ , f extends to a map from $X'$ to X by setting $f(x') := p^{-1}(\varphi (p'(x')))$ for $x'\in (p')^{-1}\varphi ^{-1}(\mathrm {Sing}\,Y)$ . By construction, $p\circ f = \varphi \circ p'$ . By this equality and the bijectivity of the map $p\colon \mathrm {Sing}\,X \to \mathrm {Sing}\,Y$ , f is continuous. Since f is holomorphic on a Zariski open subset, $f \colon X'\to X$ is holomorphic by the normality of $X'$ .

Proof. Let $X^{\natural }$ (respectively, $X'$ ) be the canonical double covering of $Y^{\natural }$ (respectively, $Y'$ ), and let $\widetilde {X}^{\natural }$ (respectively, $\widetilde {X'}$ ) be the minimal resolution of $X^{\natural }$ (respectively, $X'$ ). The birational morphism $Y'\to Y$ induces a birational morphism $\psi \colon (X',\iota ') \to (X,\iota )$ by Lemma 2.5 (3), and this $\psi $ induces an isomorphism $f\colon (\widetilde {X}',\theta ')\to (\widetilde {X},\theta )=(\widetilde {X}^{\natural },\theta )$ , by the minimality of $K3$ surfaces. Hence $(\widetilde {X}'/\theta ',(\widetilde {X}')^{\theta '})\cong (\widetilde {X}^{\natural }/\theta ,(\widetilde {X}^{\natural })^{\theta })$ . Since the projection $\widetilde {X}'/\theta '\to Y'$ (respectively, $\widetilde {X}^{\natural }/\theta \to Y^{\natural }$ ) is obtained by contracting every component of $(\widetilde {X}')^{\theta '}$ (respectively, $(\widetilde {X}^{\natural })^{\theta }$ ) to a cyclic quotient singularity of type $\frac {1}{4}(1,1)$ , f induces an isomorphism from $Y'$ to Y.

Proof. Let ${\mathcal N}_{k}$ be the isomorphism classes of good log-Enriques surfaces with k singular points (and fixed parity when $k=8$ ). By [Reference Yoshikawa43, Th. 1.8], we can identify ${\mathcal M}_{k}^{o}$ with the isomorphism classes of $2$ -elementary $K3$ surfaces of type $M_{k}$ via the period mapping. We define a map $f\colon {\mathcal N}_{k}\to {\mathcal M}_{k}^{o}$ by setting $f(Y)=(\widetilde {X},\theta )$ , where $(\widetilde {X},\theta )$ is the $2$ -elementary $K3$ surface associated to Y. Similarly, we define a map $g \colon {\mathcal M}_{k}^{o} \to {\mathcal N}_{k}$ by sending $(Z,\sigma ) \in {\mathcal M}_{k}^{o}$ to the surface obtained from $Z/\sigma $ by blowing down $Z^{\sigma }$ . Since $Z^{\sigma }$ consists of k disjoint $(-2)$ -curves, its image in $Z/\sigma $ consists of k disjoint $(-4)$ -curves so that $g(Z, \sigma )$ is a good log-Enriques surface with k singular points. Since $g=f^{-1}$ by [Reference Zhang48, Lemmas 1.4 and 2.1], f is a bijection.

Proof. Let $\omega $ be a Ricci-flat Kähler form on Z such that

(3.1) $$ \begin{align} \frac{\omega^{2}}{2!}=\eta\wedge\overline{\eta}. \end{align} $$

Since the $L^{2}$ -metric on $H^{2}(Z,{\mathcal O}_{Z})=H^{0}(Z,K_{Z})^{\lor }$ is independent of the choice of a Kähler metric on Z, we get by the anomaly formula for Quillen metrics [Reference Bismut, Gillet and Soulé6]

(3.2) $$ \begin{align} \log\left(\frac{\tau(Z,\gamma)\,\mathrm{Vol}(Z,\gamma)}{\tau(Z,\omega)\,\mathrm{Vol}(Z,\omega)}\right) = \frac{1}{24}\int_{Z}\widetilde{c_{1}c_{2}}(TZ;\gamma,\omega), \end{align} $$

where $\widetilde {c_{1}c_{2}}(TZ;\gamma ,\omega )$ is the Bott-Chern secondary class [Reference Bismut, Gillet and Soulé6] such that

$$ \begin{align*}-dd^{c}\widetilde{c_{1}c_{2}}(TZ;\gamma,\omega)=c_{1}(Z,\gamma)c_{2}(Z,\gamma)-c_{1}(Z,\omega)c_{2}(Z,\omega). \end{align*} $$

Since $c_{1}(Z,\omega )=0$ by the Ricci-flatness of $\omega $ and $\widetilde {c_{1}}(L;h,h')=\log (h/h')$ for a holomorphic line bundle L and Hermitian metrics h and $h'$ on L, and since

$$ \begin{align*}\widetilde{c_{1}c_{2}}(TZ;\gamma,\omega)=\widetilde{c_{1}}(TZ;\gamma,\omega)c_{2}(Z,\gamma)+c_{1}(Z,\omega)\widetilde{c_{2}}(TZ;\gamma,\omega) \end{align*} $$

by [Reference Gillet and Soulé21], we get by equation (3.1)

(3.3) $$ \begin{align} \widetilde{c_{1}c_{2}}(TZ;\gamma,\omega) = \widetilde{c_{1}}(TZ;\gamma,\omega)c_{2}(Z,\gamma) = \log\left(\frac{\gamma^{2}}{\omega^{2}}\right)c_{2}(Z,\gamma) = \log\left(\frac{\gamma^{2}/2!}{\eta\wedge\overline{\eta}}\right)c_{2}(Z,\gamma). \end{align} $$

Since $\mathrm {Vol}(Z,\gamma )/\mathrm {Vol}(Z,\omega )=\mathrm {Vol}(Z,\gamma )/\|\eta \|_{L^{2}}^{2}$ , we get by substituting equation (3.3) into equation (3.2)

(3.4) $$ \begin{align} \begin{aligned} \log\left(\frac{\tau(Z,\gamma)}{\tau(Z,\omega)}\right) &= -\log\left(\frac{\mathrm{Vol}(Z,\gamma)}{\|\eta\|_{L^{2}}^{2}}\right) - \frac{1}{24}\int_{Z} \log\left(\frac{\eta\wedge\overline{\eta}}{\gamma^{2}/2!}\right)\,c_{2}(Z,\gamma) \\ &= -\frac{1}{24}\int_{Z} \log\left\{\frac{\eta\wedge\overline{\eta}}{\gamma^{2}/2!}\cdot\frac{\mathrm{Vol}(Z,\gamma)}{\|\eta\|_{L^{2}}^{2}}\right\}\,c_{2}(Z,\gamma), \end{aligned} \end{align} $$

where we used the Gauss-Bonnet-Chern formula for Z to get the second equality.

Since $\omega $ is Ricci-flat, the Laplacians $\square _{0}$ and $\square _{2}$ are isospectral via the map $A^{0,0}_{Y}\ni f\mapsto f\overline {\eta }\in A^{0,2}_{Y}$ . Hence, for the Ricci-flat metric $\omega $ , we get the equality of meromorphic functions

(3.5) $$ \begin{align} \zeta_{0}(s)=\zeta_{2}(s) \end{align} $$

Since the Dolbeault complex is exact on the orthogonal complement of harmonic forms, we get the equality of meromorphic functions

(3.6) $$ \begin{align} \zeta_{0}(s)-\zeta_{1}(s)+\zeta_{2}(s)=0. \end{align} $$

By equations (3.5) and (3.6), we get

(3.7) $$ \begin{align} \tau(Z,\omega)=1. \end{align} $$

The result follows from equations (3.4) and (3.7).

Proof. See [Reference Yoshikawa43, Th. 5.7].

Proof. Let $h\in C^{\infty }(X)$ . By the definition of the Kähler form $\gamma _{\epsilon ,\delta }$ , we have

(5.1) $$ \begin{align} \begin{aligned} \int_{\widetilde{X}}\pi^{*}h\cdot c_{2}(\widetilde{X},\gamma_{\epsilon,\delta}) &= \int_{X\setminus\bigcup_{{\frak p}\in\mathrm{Sing}(X)}\widetilde{V}(\delta)_{\frak p}}h\cdot c_{2}(X,\gamma_{\epsilon,\delta}) + \sum_{{\frak p}\in\mathrm{Sing}(X)}h({\frak p})\int_{\widetilde{V}(\delta)_{\frak p}}c_{2}(\widetilde{X},\gamma_{\epsilon,\delta}) \\ &\quad+ \sum_{{\frak p}\in\mathrm{Sing}(X)}\int_{\widetilde{V}(\delta)_{\frak p}}\pi^{*}\{h-h({\frak p})\}\cdot c_{2}(\widetilde{X},\gamma_{\epsilon,\delta}). \end{aligned} \end{align} $$

For $a>0$ , let $T_{a}(z):=az$ be the homothety of $\mathbf {C}^{2}$ and let $\widetilde {T}_{a}\colon T^{*}\mathbf {P}^{1}\to T^{*}\mathbf {P}^{1}$ be the biholomorphic map induced by $T_{a}$ . Then $\widetilde {T}_{\epsilon }$ induces an isometry of Kähler manifolds

$$ \begin{align*}\widetilde{T}_{\epsilon}\colon(\widetilde{V}(\epsilon^{-2}),\epsilon^{2}\,\gamma^{\mathrm{EH}}) \cong (\widetilde{V}(1),\gamma^{\mathrm{EH}}_{\epsilon^{2}}). \end{align*} $$

Under the identification $T^{*}\mathbf {P}^{1}\setminus E\cong V(\infty )\setminus \{0\}$ , we have the following estimates

$$ \begin{align*}\left\|\gamma^{\mathrm{EH}}(z) - i\partial\bar{\partial}\|z\|^{2}\right\|\leq C(1+\|z\|)^{-4}, \qquad \|c_{2}(T^{*}\mathbf{P}^{1},\gamma^{\mathrm{EH}})(z)\|\leq C(1+\|z\|)^{-6} \end{align*} $$

for $\|z\|\gg 1$ by equation (4.3), where $C>0$ is a constant and the norm is with respect to $\gamma ^{\mathrm {EH}}$ .

Since there is a constant $C'>0$ with $\left |h|_{V(\delta )_{\frak p}}(z)-h({\frak p})\right |\leq C'\|z\|/(1+\|z\|)$ on $V(\delta )_{\frak p}$ , we get

(5.2) $$ \begin{align} \begin{aligned} \,& \left| \int_{\widetilde{V}(\delta)_{\frak p}}\{\pi^{*}h-h({\frak p})\}\cdot c_{2}(\widetilde{X},\gamma_{\epsilon,\delta}) \right| = \left| \int_{\widetilde{V}(\delta)_{\frak p}} \pi^{*}\{h|_{V(\delta)_{\frak p}}-h({\frak p})\}\cdot c_{2}(T^{*}\mathbf{P}^{1},\gamma^{\mathrm{EH}}_{\epsilon}) \right| \\ &= \left| \int_{\widetilde{V}(\delta\sqrt{\epsilon}^{-1})}\widetilde{T}_{\epsilon}^{*} \pi^{*}\{h|_{V(\delta)_{\frak p}}-h({\frak p})\}\cdot c_{2}(T^{*}\mathbf{P}^{1},\gamma^{\mathrm{EH}}) \right| \\ &\leq \int_{V(\delta\sqrt{\epsilon}^{-1})}C'\frac{\sqrt{\epsilon}\|z\|}{1+\sqrt{\epsilon}\|z\|}\cdot\frac{C}{1+\|z\|^{6}}\,\frac{(\gamma^{\mathrm{EH}})^{2}}{2!} \leq C"\sqrt{\epsilon}\to0\quad(\epsilon\to0), \end{aligned} \end{align} $$

where $C^{\prime \prime }>0$ is a constant. By equations (5.1) and (5.2), we get

(5.3) $$ \begin{align} \begin{aligned} \lim_{\epsilon\to0}\int_{\widetilde{X}}\pi^{*}h\cdot c_{2}(\widetilde{X},\gamma_{\epsilon,\delta}) &= \int_{X\setminus\bigcup_{{\frak p}\in\mathrm{Sing}(X)}V(\delta)_{\frak p}}h\cdot c_{2}(X,\gamma_{0}) \\ &\quad + \sum_{{\frak p}\in\mathrm{Sing}(X)}h({\frak p})\int_{T^{*}\mathbf{P}^{1}}c_{2}(T^{*}\mathbf{P}^{1},\gamma^{\mathrm{EH}}) \\ &= \int_{X}h\cdot c_{2}(X,\gamma_{0}) + \sum_{{\frak p}\in\mathrm{Sing}(X)}h({\frak p})\int_{T^{*}\mathbf{P}^{1}}c_{2}(T^{*}\mathbf{P}^{1},\gamma^{\mathrm{EH}}), \end{aligned} \end{align} $$

where we used the vanishing of $c_{2}(X,\gamma _{0})$ on $V(\delta )_{\frak p}$ to get the second equality. Setting $h=1$ in equation (5.3) and comparing it with the formula [Reference Kobayash27, p.396 l.5], we get

(5.4) $$ \begin{align} \int_{T^{*}\mathbf{P}^{1}}c_{2}(T^{*}\mathbf{P}^{1},\gamma^{\mathrm{EH}})=\chi(\mathbf{P}^{1})-\frac{1}{|\mathbf{Z}_{2}|}=\frac{3}{2}. \end{align} $$

The first assertion follows from equations (5.3) and (5.4).

Since $\#\mathrm {Sing}(Y)=k$ , we get by the first assertion

$$ \begin{align*}2\int_{Y}c_{2}(Y,\gamma_{0}) = \int_{X}c_{2}(X,\gamma_{0}) = \int_{\widetilde{X}}c_{2}(\widetilde{X},\gamma_{\epsilon,\delta})-\frac{3}{2}\sum_{{\frak p}\in\mathrm{Sing}(X)}\int_{X}\delta_{\frak p} = 24-\frac{3}{2}k. \end{align*} $$

This proves the second assertion.

Proof. Since $\gamma _{\epsilon ,\delta }$ converges to $\gamma _{0}$ outside $\bigcup _{{\frak p}\in \mathrm {Sing}(X)}\widetilde {V}(\delta )_{\frak p}$ , and since $\mathrm {Vol}(\widetilde {X},\gamma _{\epsilon ,\delta })$ converges to $\mathrm {Vol}(X,\gamma _{0})$ as $\epsilon \to 0$ , we get the convergence

$$ \begin{align*}\begin{aligned} \,& \left( \int_{\widetilde{X}\setminus\bigcup_{{\frak p}\in\mathrm{Sing}(X)}\widetilde{V}(\delta)_{\frak p}} + \sum_{{\frak p}\in\mathrm{Sing}(X)}\int_{\widetilde{V}(\delta)_{\frak p}} \right) \log \left\{ \frac{\eta\wedge\overline{\eta}}{\gamma_{\epsilon,\delta}^{2}/2!}\cdot\frac{\mathrm{Vol}(\widetilde{X},\gamma_{\epsilon,\delta})}{\|\eta\|_{L^{2}}^{2}} \right\} c_{2}(\widetilde{X},\gamma_{\epsilon,\delta}) \\ &\to \int_{\widetilde{X}\setminus\bigcup_{{\frak p}\in\mathrm{Sing}(X)}\widetilde{V}(\delta)_{\frak p}} \log \left\{ \frac{\eta\wedge\overline{\eta}}{\gamma_{0}^{2}/2!}\cdot\frac{\mathrm{Vol}(X,\gamma_{0})}{\|\eta\|_{L^{2}}^{2}} \right\} c_{2}(\widetilde{X},\gamma_{0}) \\ &\quad+ \lim_{\epsilon\to0} \sum_{{\frak p}\in\mathrm{Sing}(X)}\int_{\widetilde{V}(\delta)_{\frak p}} \left\{ \log\pi^{*}|f_{\frak p}(z)|^{2}c_{2}(\widetilde{X},\gamma_{\epsilon,\delta}) + \log\frac{\mathrm{Vol}(\widetilde{X},\gamma_{\epsilon,\delta})}{\|\eta\|_{L^{2}}^{2}}c_{2}(\widetilde{X},\gamma_{\epsilon,\delta}) \right\} \\ &= \int_{X\setminus\bigcup_{{\frak p}\in\mathrm{Sing}(X)}V(\delta)_{\frak p}} \log \left\{ \frac{\eta\wedge\overline{\eta}}{\gamma_{0}^{2}/2!}\cdot\frac{\mathrm{Vol}(X,\gamma_{0})}{\|\eta\|_{L^{2}}^{2}} \right\} c_{2}(X,\gamma_{0}) \\ &\quad+ \frac{3}{2}\sum_{{\frak p}\in\mathrm{Sing}(X)}\log \left(|f_{\frak p}(0)|^{2}\frac{\mathrm{Vol}(X,\gamma_{0})}{\|\eta\|_{L^{2}}^{2}}\right) \end{aligned} \end{align*} $$

as $\epsilon \to 0$ , where the last equality follows from Proposition 5.1. Since $c_{2}(X,\gamma _{0})=0$ on $\bigcup _{{\frak p}\in \mathrm {Sing}(X)}V(\delta )_{\frak p}$ , we get the result.

Proof. By Theorem 3.2 and Proposition 5.2, we get the desired equality. The independence of the double limit $\lim _{\delta \to 0}\lim _{\epsilon \to 0}\tau (\widetilde {X},\gamma _{\epsilon ,\delta })$ from $\rho $ is obvious because the right-hand side is independent of the choice of $\rho $ .

Proof. Since $\gamma _{\epsilon ,\delta }|_{E_{\frak p}}=\epsilon \,\omega _{\mathrm {FS}}(E_{\frak p})$ , and since $\omega _{\mathrm {FS}}(E_{\frak p})$ is Kähler-Einstein, we get

$$ \begin{align*}c_{1}(\widetilde{X}^{\theta},\gamma_{\epsilon,\delta}|_{\widetilde{X}^{\theta}})|_{E_{\frak p}} = \chi(\mathbf{P}^{1})\,\omega_{\mathrm{FS}}(E_{\frak p}) = 2\,\omega_{\mathrm{FS}}(E_{\frak p}). \end{align*} $$

Since $(\eta \wedge \overline {\eta })/(\gamma _{\epsilon ,\delta }^{2}/2!)|_{E_{\frak p}}=|f_{\frak p}(0)|^{2}$ by equation (4.13), we get

$$ \begin{align*}\begin{aligned} \,& A_{M}(\widetilde{X},\theta,\gamma_{\epsilon,\delta}) = \exp \left[ \frac{1}{8}\int_{\widetilde{X}^{\theta}} \left.\log\left\{\frac{\eta\wedge\overline{\eta}}{\gamma_{\epsilon,\delta}^{2}/2!} \cdot \frac{\mathrm{Vol}(\widetilde{X},\gamma_{\epsilon,\delta})}{\|\eta\|_{L^{2}}^{2}}\right\}\right|_{\widetilde{X}^{\theta}}\, c_{1}(\widetilde{X}^{\theta},\gamma_{\epsilon,\delta}|_{\widetilde{X}^{\theta}}) \right] \\ &= \exp \left[ \frac{1}{4}\sum_{{\frak p}\in\mathrm{Sing}(X)}\int_{E_{\frak p}} \left.\log\left\{ |f_{\frak p}(0)|^{2}\frac{\mathrm{Vol}(\widetilde{X},\gamma_{\epsilon,\delta})}{\|\eta\|_{L^{2}}^{2}}\right\}\right|_{E_{\frak p}}\, \omega_{\mathrm{FS}}(E_{\frak p}) \right] \\ &= \exp \left[ \frac{1}{4}\sum_{{\frak p}\in\mathrm{Sing}(X)} \log\left\{ |f_{\frak p}(0)|^{2}\frac{\mathrm{Vol}(\widetilde{X},\gamma_{\epsilon,\delta})}{\|\eta\|_{L^{2}}^{2}}\right\} \right] \to \prod_{{\frak p}\in\mathrm{Sing}(X)}\left\{|f_{\frak p}(0)|^{2}\frac{\mathrm{Vol}(X,\gamma_{0})}{\|\eta\|_{L^{2}}^{2}}\right\}^{\frac{1}{4}} \end{aligned} \end{align*} $$

as $\epsilon \to 0$ , where we used equation (4.14) to get the last limit. This completes the proof.

Proof. We recall a formula of Bost [Reference Bost9, Prop. 4.4]. Let $(Z,g)$ be a compact Kähler manifold of dimension d, and let $\lambda>0$ be a constant. By [Reference Bost9, (4.2.4)], we get

(5.5) $$ \begin{align} \log\left(\frac{\tau(Z,\lambda g)}{\tau(Z,g)}\right) = \left( -\sum_{i=0}^{d}(-1)^{i}(d-i)\,h^{0,i}(Z)+\int_{Z}\mathrm{Td}'(TZ) \right) \log\lambda, \end{align} $$

where the characteristic class $\mathrm {Td}'(E)$ is defined as follows (compare [Reference Bost9, Prop. 4.4]). If $\xi _{i} (i=1,\ldots ,r=\mathrm {rk}(E))$ are the Chern roots of a vector bundle E, then

$$ \begin{align*}\mathrm{Td}'(E) := \mathrm{Td}(E)\cdot\sum_{i=1}^{r}\left(\frac{1}{\xi_{i}}-\frac{e^{-\xi_{i}}}{1-e^{-\xi_{i}}}\right). \end{align*} $$

Since

$$ \begin{align*}\mathrm{Td}'(x) = \frac{x}{1-e^{-x}}\left(\frac{1}{x}-\frac{e^{-x}}{1-e^{-x}}\right) = \frac{1}{2}+\frac{1}{6}x+O(x^{2}) \end{align*} $$

and hence $\int _{\mathbf {P}^{1}}\mathrm {Td}'(T\mathbf {P}^{1})=1/3$ , we get by equation (5.5) applied to $(Z,g)=(\mathbf {P}^{1},\omega _{\mathrm {FS}})$

(5.6) $$ \begin{align} \tau(E_{\frak p},\gamma_{\epsilon,\delta}|_{E_{\frak p}})/\tau(\mathbf{P}^{1},\omega_{\mathrm{FS}}) = \tau(\mathbf{P}^{1},\epsilon\,\omega_{\mathrm{FS}})/\tau(\mathbf{P}^{1},\omega_{\mathrm{FS}}) = \epsilon^{-2/3}. \end{align} $$

Since

(5.7) $$ \begin{align} \mathrm{Vol}(E_{\frak p},\gamma_{\epsilon,\delta}|_{E_{\frak p}})/\mathrm{Vol}(\mathbf{P}^{1},\omega_{\mathrm{FS}}) = \mathrm{Vol}(\mathbf{P}^{1},\epsilon\,\omega_{\mathrm{FS}})/\mathrm{Vol}(\mathbf{P}^{1},\omega_{\mathrm{FS}}) = \epsilon, \end{align} $$

the result follows from equations (5.6) and (5.7).

Proof. Since $(\widetilde {V}(\infty ),\gamma ^{\mathrm {EH}}_{\epsilon })\cong (\widetilde {V}(\infty ),\epsilon \gamma ^{\mathrm {EH}})$ by equation (4.1), and since the inequalities (1), (2), (3) above are invariant under the scaling of metrics $\gamma ^{\mathrm {EH}}\mapsto \epsilon \gamma ^{\mathrm {EH}}$ , it suffices to prove (1), (2), (3) for $\gamma ^{\mathrm {EH}}$ . In the rest of the proof, all norms are defined with respect to $\gamma ^{\mathrm {EH}}$ . Identifying a function in $C_0^{\infty }(\widetilde {V}(\infty )\! -\! K)$ with the corresponding $\pm 1$ -invariant function on $\mathbf R^4$ with compact support via $\Phi $ , we deduce from the Sobolev inequality for $\mathbf R^4$ that

$$ \begin{align*}\|f\|_{L^4(\widetilde{V}(\infty))} \leq 2C\|df\|_{L^2(\widetilde{V}(\infty))}, \quad \forall f \in C_0^{\infty}(\widetilde{V}(\infty)\!-\! K),\end{align*} $$

where C is the Sobolev constant for $\mathbf R^4$ . By an argument using partition of unity, there is a constant $C_K> 0$ such that

$$ \begin{align*}\|f\|_{L^4(\widetilde{V}(\infty))} \leq C_K(\|df\|_{L^2(\widetilde{V}(\infty))} + \|f\|_{L^2(K)}), \quad \forall f \in C_0^{\infty}(\widetilde{V}(\infty)).\end{align*} $$

Assume that there is no constant $D> 0$ such that

$$ \begin{align*}\|f\|_{L^2(K)}\leq D\|df\|_{L^2(\widetilde{V}(\infty))}, \quad \forall f \in C_0^{\infty}(\widetilde{V}(\infty)).\end{align*} $$

Then for any $n \in \mathbf N$ , there is a function $f_n \in C_0^{\infty }(\widetilde {V}(\infty ))$ such that

$$ \begin{align*}\|f_n\|_{L^2(K)} = 1, \quad \|df_n\|_{L^2(\widetilde{V}(\infty))} \leq \frac{1}{n}.\end{align*} $$

Therefore, we have

$$ \begin{align*}\|f_n\|_{L^4(\widetilde{V}(\infty))}\leq C_K(1 + 1/n) \leq 2C_K.\end{align*} $$

Passing to a subsequence if necessary, it follows that the sequence $f_n$ has a weak limit $f_{\infty } \in L^4(\widetilde {V}(\infty ))$ with $df_{\infty }= 0$ as currents on $\widetilde {V}(\infty )$ . This implies that in $L^4(\widetilde {V}(\infty ))$ , $f_{\infty } = 0$ . On the other hand, let $K'$ be a sufficiently big compact subset of $\widetilde {V}(\infty )$ , whose open subset contains K. Now, for any compact subset $K' \subset \widetilde {V}(\infty )$ , there is a constant $C_{K'}> 0$ such that

$$ \begin{align*}\|f_n\|_{L^2(K')}\leq \mathrm{Vol}(K')^{1/2}\|f_n\|^{1/2}_{L^4(K')}\leq C_{K'} =\sqrt{2C_K\mathrm{Vol}(K')}.\end{align*} $$

Hence, by the Rellich lemma, we may assume (by again passing to a subsequence if necessary) that $f_n$ converges to $f_{\infty }$ strongly in $L^2(K')$ . Since $K \subset K'$ and hence the convergence $f_n \rightarrow f_{\infty }$ in $L^2(K)$ is strong, we see that $ \|f_{\infty }\|_{L^2(K)} = \lim _{n \rightarrow \infty } \|f_n\|_{L^2(K)} = 1$ . This is a contradiction. Hence there exists a constant D such that $ \|f\|_{L^2(K)}\leq D\|df\|_{L^2(\widetilde {V}(\infty ))}$ . By setting $C = CK(1 + D$ ), we have

$$ \begin{align*}\| f \|_{L^4(\widetilde{V}(\infty))} \leq C \| df \|_{L^2(\widetilde{V}(\infty))}.\end{align*} $$

This proves (1).

(2) is an immediate consequence of (1), and the isomorphism $C_0^{\infty }(\widetilde {V}(\infty ))\ni f\mapsto f\overline {\eta }\in A^{0,2}_{0}(\widetilde {V}(\infty ))$ , which commutes with the operations involved. To see (3), let $\alpha \in A^{0,1}_0(\widetilde {V}(\infty ))$ . Then by (1),

$$ \begin{align*}\| \alpha \|^2_{L^4(\widetilde{V}(\infty))} = \left( \int_{\widetilde{V}(\infty)} |\alpha|^4 dx\right)^{1/2} \leq C^2 \int_{\widetilde{V}(\infty)} |\, d|\alpha|\, |^2dx. \end{align*} $$

Using Kato’s inequality, we have

$$ \begin{align*}\int_{\widetilde{V}(\infty)} |\, d|\alpha|\, |^2dx \leq \int_{\widetilde{V}(\infty)} |\nabla\alpha|^2dx.\end{align*} $$

Now the Bochner formula [Reference Ma and Marinescu34, (1.4.63)] gives $(\overline {\partial }\overline {\partial }^* + \overline {\partial }^*\overline {\partial })\alpha = \nabla ^*\nabla \alpha $ since $(\widetilde {V}(\infty ),\gamma ^{\mathrm {EH}}_{\epsilon })$ is Ricci flat. Our result follows.

Proof. By Lemmas 4.1 and 4.3, there exist constants $C_{1},C_{2}>0$ such that

(6.1) $$ \begin{align} C_{1}\gamma^{\mathrm{EH}}_{\epsilon}\leq\kappa_{\epsilon,\delta}\leq C_{2}\gamma^{\mathrm{EH}}_{\epsilon} \end{align} $$

for all $\epsilon ,\delta \in (0,1]$ with $\epsilon \delta ^{-2}\leq \epsilon (\rho )$ . Here $\epsilon (\rho )$ is defined in Section 4.2.1, in the discussion following equation (4.2). Hence there is a constant $C_{3}>0$ such that

(6.2) $$ \begin{align} C_{3}^{-1}\| \alpha \|^2_{L^4(\widetilde{V}(\infty),\gamma^{\mathrm{EH}}_{\epsilon})} \leq \| \alpha \|^2_{L^4(\widetilde{V}(\infty),\kappa_{\epsilon,\delta})} \leq C_{3}\| \alpha \|^2_{L^4(\widetilde{V}(\infty),\gamma^{\mathrm{EH}}_{\epsilon})}, \end{align} $$

(6.3) $$ \begin{align} C_{3}^{-1}\| \overline{\partial}\alpha \|^2_{L^{2}(\widetilde{V}(\infty),\gamma^{\mathrm{EH}}_{\epsilon})} \leq \| \overline{\partial}\alpha \|^2_{L^{2}(\widetilde{V}(\infty),\kappa_{\epsilon,\delta})} \leq C_{3}\| \overline{\partial}\alpha \|^2_{L^{2}(\widetilde{V}(\infty),\gamma^{\mathrm{EH}}_{\epsilon})} \end{align} $$

for all $\epsilon ,\delta \in (0,1]$ with $\epsilon \delta ^{-2}\leq \epsilon (\rho )$ and $\alpha \in A^{0,q}_{0}(\widetilde {V}(\infty ))$ .

Let $\Lambda _{\epsilon .\delta }$ (respectively, $\Lambda _{\epsilon }$ ) be the Lefschetz operator defined as the adjoint of the multiplication by $\kappa _{\epsilon ,\delta }$ (respectively, $\gamma ^{\mathrm {EH}}_{\epsilon }$ ). Since $\overline {\partial }^{*}=\pm i\Lambda _{\epsilon ,\delta }\partial $ for $(0,q)$ -forms by the Kähler identity, there exists by equation (6.1) a constant $C_{4}>0$ such that

(6.4) $$ \begin{align} C_{4}^{-1}\| \overline{\partial}^*\alpha \|^2_{L^2(\widetilde{V}(\infty),\gamma^{\mathrm{EH}}_{\epsilon})} \leq \| \overline{\partial}^*\alpha \|^2_{L^2(\widetilde{V}(\infty),\kappa_{\epsilon,\delta})} \leq C_{4}\| \overline{\partial}^*\alpha \|^2_{L^2(\widetilde{V}(\infty),\gamma^{\mathrm{EH}}_{\epsilon})}. \end{align} $$

By Lemma 6.1 (3) and equations (6.2), (6.3) and (6.4), we get the result.

Proof. Since $\gamma _{\epsilon ,\delta }=\kappa _{\epsilon ,\delta }$ on $\bigcup _{{\frak p}\in \mathrm {Sing}(X)}\widetilde {V}(\delta )_{\frak p}$ , the result follows from Lemma 6.2 and an easy partition of unity argument.

Proof. (Case 1) Let $q=0$ . By Proposition 6.3, the Sobolev constant is uniform for $\epsilon ,\delta \in (0,1]$ with $\epsilon \delta ^{-2}\leq \epsilon (\rho )$ . By [Reference Carlen, Kusuoka and Stroock10, Thms. 2.1 and 2.16], there are constants $A>0$ , $B\geq 0$ such that for all $\epsilon ,\delta \in (0,1]$ with $\epsilon \delta ^{-2}\leq \epsilon (\rho )$ , and $x,y\in \widetilde {X}$ , $t>0$ ,

(6.8) $$ \begin{align} 0< K_{\epsilon,\delta}^{q}(t,x,y) \leq A\,e^{Bt}t^{-2}. \end{align} $$

Let $q=2$ . By Lemma 4.5, the Lichnerowicz formula and [Reference Hess, Schrader and Uhlenbrock24, p.32 l.4-l.5], we have

(6.9) $$ \begin{align} | K_{\epsilon,\delta}^{q}(t,x,y) | \leq e^{t|\mathrm{Ric}\gamma_{\epsilon,\delta}|_{\infty}} | K_{\epsilon,\delta}^{0}(t,x,y) | \leq e^{C(\epsilon\delta^{-4}+1)t}Ae^{Bt}(t^{-2}+1). \end{align} $$

For $t\leq 1$ , we get equation (6.6) by equations (6.8) and (6.9). For $t\geq 1$ , since $\mathrm {tr}\,K_{\epsilon ,\delta }^{q}(t,x,x)$ is a decreasing function in t, we deduce equation (6.6) from equations (6.5), (6.8) and (6.9) and the inequality

$$ \begin{align*}\left| K_{\epsilon,\delta}^{q}(t,x,y) \right| \leq \sqrt{\mathrm{tr}\,K_{\epsilon,\delta}^{q}(1,x,x)}\sqrt{\mathrm{tr}\,K_{\epsilon,\delta}^{q}(1,y,y)} \leq 2e^{C(\epsilon\delta^{-4}+1)}Ae^{B}. \end{align*} $$

Since $\mathrm {Tr}\,e^{-t\,\square _{\epsilon ,\delta }^{q}}=\int _{\widetilde {X}}\mathrm {tr}\,K_{\epsilon ,\delta }^{q}(t,x,x)\,dx$ , we get equation (6.7) from equation (6.6).

(Case 2) Let $q=1$ . Since $\sum _{q}(-1)^{q}\mathrm {Tr}\,e^{-t\,\square _{\epsilon ,\delta }^{q}}=0$ for all $t>0$ , equation (6.7) for $q=1$ follows from equation (6.7) for $q=0,2$ . This completes the proof.

Proof. Since $\dim \widetilde {X}=2$ and hence $\lambda _{\epsilon }^{1}(1)=\lambda _{\epsilon }^{0}(1)$ or $\lambda _{\epsilon }^{1}(1)=\lambda _{\epsilon }^{2}(1)$ , it suffices to prove equation (6.10) for $q=0,2$ . Assume that there is a sequence $\{\epsilon _{n}\}$ such that $\epsilon _{n}\to 0$ and $\lambda _{\epsilon _{n}}^{q}(1)\to 0$ as $n\to \infty $ for $q=0$ or $2$ . By the same argument as in [Reference Yoshikawa42, p.434–p.436] using the uniformity of the Sobolev constant (compare Proposition 6.3), there is a holomorphic q-form $\psi $ on $X \setminus \mathrm {Sing}\,X$ , which is possibly meromorphic on $\widetilde {X}$ , with the following properties:

1. (i) The complex conjugation $\overline {\varphi _{1,\epsilon _{n}}^{q}}$ converges to $\psi $ on every compact subset of $X\setminus \mathrm {Sing}(X)$ as $n\to \infty $ .

2. (ii) $\|\psi \|_{L^{2}}=1$ and $\pi ^{*}\psi \perp H^{0}(\widetilde {X},\Omega ^{q}_{\widetilde {X}})$ with respect to the degenerate Kähler metric $\pi ^{*}\gamma _{0}$ on $\widetilde {X}$ .

Since $\mathrm {Sing}\,X$ consists of isolated orbifold points, it follows from the Riemann extension theorem that $\psi $ extends to a holomorphic q-form on X in the sense of orbifolds. When $q=0$ , $\psi $ is a constant. When $q=2$ , since X has canonical singularities, $\pi ^{*}\psi $ is a holomorphic $2$ -form on $\widetilde {X}$ . In both cases, the condition $\pi ^{*}\psi \perp H^{0}(\widetilde {X},\Omega ^{q}_{\widetilde {X}})$ implies $\psi =0$ , which contradicts the other condition $\|\psi \|_{L^{2}}=1$ . This proves the result.

Proof. Firstly, we prove the inequality when $q=1$ . Since $\widetilde {X}$ is a $K3$ surface and hence $\ker \square _{\epsilon ,\delta }^{1}=0$ , we get by equation (6.10)

(6.11) $$ \begin{align} \lambda\,\|\alpha\|_{L^{2}(\widetilde{X},\gamma_{\epsilon})}^{2} \leq \|\bar{\partial}\alpha\|_{L^{2}(\widetilde{X},\gamma_{\epsilon})}^{2}+\|\bar{\partial}^{*}\alpha\|_{L^{2}(\widetilde{X},\gamma_{\epsilon})}^{2} = \|\partial\alpha\|_{L^{2}(\widetilde{X},\gamma_{\epsilon})}^{2} \end{align} $$

for all $\alpha \in A^{0,1}(\widetilde {X})$ , where we used the coincidence of the $\bar {\partial }$ -Laplacian and the $\partial $ -Laplacian for Kähler manifolds to get the equality in equation (6.11). By Lemma 4.4, there exist constants $C_{1}>0$ such that for all $\alpha \in A^{0,1}(\widetilde {X})$ ,

$$ \begin{align*}C_{1}^{-1}\|\alpha\|_{L^{2}(\widetilde{X},\gamma_{\epsilon})}^{2} \leq \|\alpha\|_{L^{2}(\widetilde{X},\gamma_{\epsilon,\delta})}^{2} \leq C_{1}\|\alpha\|_{L^{2}(\widetilde{X},\gamma_{\epsilon})}^{2}, \end{align*} $$

$$ \begin{align*}C_{1}^{-1}\|\partial\alpha\|_{L^{2}(\widetilde{X},\gamma_{\epsilon})}^{2} \leq \|\partial\alpha\|_{L^{2}(\widetilde{X},\gamma_{\epsilon,\delta})}^{2} \leq C_{1}\|\partial\alpha\|_{L^{2}(\widetilde{X},\gamma_{\epsilon})}^{2}. \end{align*} $$

Combining these inequalities and equation (6.11), we get for all $\alpha \in A^{0,1}(\widetilde {X})$

(6.12) $$ \begin{align} C_{1}^{-1}\lambda\,\|\alpha\|_{L^{2}(\widetilde{X},\gamma_{\epsilon,\delta})}^{2} \leq C_{1}\|\partial\alpha\|_{L^{2}(\widetilde{X},\gamma_{\epsilon,\delta})}^{2} = C_{1} \left( \|\bar{\partial}\alpha\|_{L^{2}(\widetilde{X},\gamma_{\epsilon,\delta})}^{2}+\|\bar{\partial}^{*}\alpha\|_{L^{2}(\widetilde{X},\gamma_{\epsilon,\delta})}^{2} \right). \end{align} $$

The result for $q=1$ follows from equation (6.12). Since $\bar {\partial }\varphi _{\epsilon ,\delta }^{0}(1)$ and $\bar {\partial }^{*}\varphi _{\epsilon ,\delta }^{2}(1)$ are nonzero eigenforms of $\square _{\epsilon ,\delta }^{1}$ , we get $\lambda '\leq \lambda _{\epsilon ,\delta }^{1}(1)\leq \lambda _{\epsilon ,\delta }^{0}(1)$ and $\lambda '\leq \lambda _{\epsilon ,\delta }^{1}(1)\leq \lambda _{\epsilon ,\delta }^{2}(1)$ .

Proof. By Proposition 6.4, we get for all $\epsilon ,\delta \in (0,1]$ with $\epsilon \delta ^{-2}\leq \epsilon (\rho )$ and $t\in (0,1]$

$$ \begin{align*}\sum_{i=1}^{k}e^{-t\lambda_{\epsilon,\delta}^{q}(i)} \leq h^{0,q}(\widetilde{X})+\sum_{i=1}^{\infty}e^{-t\lambda_{\epsilon,\delta}^{q}(i)} = \mathrm{Tr}\,e^{-t\square_{\epsilon,\delta}^{q}} \leq A'e^{C(\epsilon\delta^{-4}+1)} \,t^{-2}, \end{align*} $$

where $A'$ is a constant such that $A\mathrm {Vol}(\widetilde {X},\gamma _{\epsilon ,\delta }) \leq A'$ . Since $\lambda '/\lambda _{\epsilon ,\delta }^{q}(k)\leq 1$ by Lemma 6.6, substituting $t:=\lambda '/\lambda _{\epsilon ,\delta }^{q}(k)$ in this inequality and using $\lambda _{\epsilon ,\delta }^{q}(i)/\lambda _{\epsilon ,\delta }^{q}(k)\leq 1$ for $i\leq k$ , we get

$$ \begin{align*}k\,e^{-\lambda'} \leq \sum_{i=1}^{k}e^{-\frac{\lambda'\lambda_{\epsilon,\delta}^{q}(i)}{\lambda_{\epsilon,\delta}^{q}(k)}} \leq A'e^{C(\epsilon\delta^{-4}+1)}\,\left(\frac{\lambda'}{\lambda_{\epsilon,\delta}^{q}(k)}\right)^{-2}. \end{align*} $$

We get the result by setting $\Lambda :=(A')^{-1/2}\lambda ' e^{-\lambda '/2}$ .

Proof. Since $\lambda _{\epsilon ,\delta }^{q}(k)\geq \frac {\Lambda (\epsilon \delta ^{-4}+1)}{2}(k^{1/2}+1)$ by Theorem 6.7, we get $\sum _{k=1}^{\infty }e^{-t\lambda _{\epsilon ,\delta }^{q}(k)} \leq e^{-t\Lambda (\epsilon \delta ^{-4}+1)/2}\sum _{k=1}^{\infty }e^{-\frac {1}{2}t\Lambda (\epsilon \delta ^{-4}+1) k^{1/2}} \leq \Psi (\epsilon \delta ^{-4}+1)\,e^{-t\Lambda (\epsilon \delta ^{-4}+1)/2}$ for $t\geq 1$ .

Proof. When $q=0$ , the result follows from Lemma 6.2 and [Reference Carlen, Kusuoka and Stroock10, Thms. 2.1 and 2.16]. Let $q>0$ . Since $|\mathrm {Ric}(\gamma _{\epsilon ,\delta })|\leq C(\epsilon \delta ^{-4}+1)$ by Lemma 4.5, we deduce from [Reference Hess, Schrader and Uhlenbrock24, p.32 l.4-l.5] and the Lichnerowicz formula for $\square _{\epsilon ,\delta }^{q}$ that

$$ \begin{align*}0< \left| K_{\epsilon,\delta,\infty}^{q}(t,x,y) \right| \leq e^{C(\epsilon\delta^{-4}+1)t}K_{\epsilon,\delta,\infty}^{0}(t,x,y) \leq A\,e^{C(\epsilon\delta^{-4}+1)t}t^{-2}. \end{align*} $$

This proves the result for $t\leq 1$ . Since equation (6.5) remains valid for $K_{\epsilon ,\delta ,\infty }^{q}(t,x,y)$ by the fact that $K_{\epsilon ,\delta ,\infty }^{q}(t,x,y)$ is obtained as the limit $R\to \infty $ of the Dirichlet heat kernel of $\widetilde {V}(R)$ , the result for $t\geq 1$ also follows. This completes the proof.

Proof. (Step 1) Let $g_{0},g_{1}$ be $\theta $ -invariant Kähler metrics on $\widetilde {X}$ . Let $\widetilde {\mathrm {Td}}_{\theta }(T\widetilde {X};g_{0},g_{1})^{(1,1)}$ be the Bott-Chern class such that

$$ \begin{align*}-dd^{c}\widetilde{\mathrm{Td}}_{\theta}(T\widetilde{X};g_{0},g_{1}) = \mathrm{Td}_{\theta}(T\widetilde{X},g_{0}) - \mathrm{Td}_{\theta}(T\widetilde{X},g_{1}). \end{align*} $$

By Bismut [Reference Bismut4, Th. 2.5],

(7.1) $$ \begin{align} \log \left( \frac{\tau_{\mathbf{Z}_{2}}(\widetilde{X},g_{0})(\theta)\mathrm{Vol}(\widetilde{X},g_{0})}{\tau_{\mathbf{Z}_{2}}(\widetilde{X},g_{1})(\theta)\mathrm{ Vol}(\widetilde{X},g_{1})} \right) = \int_{\widetilde{X}^{\theta}} \widetilde{\mathrm{Td}}_{\theta}(T\widetilde{X};g_{0},g_{1}). \end{align} $$

Since

$$ \begin{align*}\mathrm{Td}_{\theta}(T\widetilde{X};g_{0},g_{1})^{(1,1)} = \frac{1}{8} {c_{1}(T\widetilde{X})|_{\widetilde{X}^{\theta}}c_{1}(T\widetilde{X}^{\theta})}(g_{0},g_{1}) - \frac{1}{12} {c_{1}(T\widetilde{X}^{\theta})^{2}}(g_{0},g_{1}) \end{align*} $$

by [Reference Yoshikawa43, Prop. 5.3], we have the following equality of Bott-Chern classes:

$$ \begin{align*}\begin{aligned} \widetilde{\mathrm{Td}}_{\theta}(T\widetilde{X};g_{0},g_{1})^{(1,1)} &= \frac{1}{8} \widetilde{c_{1}(T\widetilde{X})|_{\widetilde{X}^{\theta}}c_{1}(T\widetilde{X}^{\theta})}(g_{0},g_{1}) - \frac{1}{12} \widetilde{c_{1}(T\widetilde{X}^{\theta})^{2}}(g_{0},g_{1}) \\ &= \frac{1}{8} \widetilde{c_{1}}(T\widetilde{X};g_{0},g_{1})|_{\widetilde{X}^{\theta}}c_{1}(T\widetilde{X}^{\theta},g_{1}) + \frac{1}{8} c_{1}(T\widetilde{X},g_{0})|_{\widetilde{X}^{\theta}}\widetilde{c_{1}}(T\widetilde{X}^{\theta};g_{0},g_{1}) \\ &\quad - \frac{1}{12} \widetilde{c_{1}}(T\widetilde{X}^{\theta};g_{0},g_{1})\{c_{1}(T\widetilde{X}^{\theta},g_{0})+c_{1}(T\widetilde{X}^{\theta},g_{1})\}, \end{aligned} \end{align*} $$

where [Reference Gillet and Soulé21, Eq. (1.3.1.2)] is used to get the second equality. For a holomorphic line bundle L and Hermitian metrics $h_{0},h_{1}$ on L, we have

$$ \begin{align*}\widetilde{c_{1}}(L;h_{0},h_{1})=\log(h_{0}/h_{1}) \end{align*} $$

by [Reference Gillet and Soulé21, Eq. (1.2.5.1)]. (Our sign convention is different from the one in Gillet-Soulé [Reference Gillet and Soulé21]. Our $\widetilde {c_{1}}(L;h_{0},h_{1})$ is $-\widetilde {c_{1}}(L;h_{0},h_{1})$ in [Reference Gillet and Soulé21].) Hence

(7.2) $$ \begin{align} \begin{aligned} \widetilde{\mathrm{Td}}_{\theta}(T\widetilde{X};g_{0},g_{1})^{(1,1)} &\equiv \frac{1}{8} \left. \log\left(\frac{\det g_{0}}{\det g_{1}}\right) \right|_{\widetilde{X}^{\theta}} c_{1}(\widetilde{X}^{\theta},g_{1}) + \frac{1}{8} \left. c_{1}(\widetilde{X},g_{0}) \log\left(\frac{g_{0}}{g_{1}} \right|_{\widetilde{X}^{\theta}} \right) \\ &\quad - \frac{1}{12} \left. \log\left(\frac{g_{0}}{g_{1}} \right|_{\widetilde{X}^{\theta}} \right) \{c_{1}(\widetilde{X}^{\theta},g_{0})+c_{1}(\widetilde{X}^{\theta},g_{1})\} \\ &\qquad \mod \mathrm{Im}\,\partial+\mathrm{Im}\,\bar{\partial}. \end{aligned} \end{align} $$

(Step 2) We set $g_{0}=\gamma _{\epsilon ,\delta }$ and $g_{1}=\gamma _{\epsilon (\rho )}$ in Step 1. Since $g_{0}=\gamma _{\epsilon ,\delta }$ is Ricci-flat on a neighbourhood of $\widetilde {X}^{\theta }$ , we have

$$ \begin{align*}c_{1}(\widetilde{X},\gamma_{\epsilon,\delta})|_{\widetilde{X}^{\theta}}=0. \end{align*} $$

Since the volume form of EH instanton $i\partial \bar {\partial }F_{\epsilon }$ is the standard Euclidean volume form

$$ \begin{align*}\frac{(i\partial\bar{\partial}F_{\epsilon})^{2}}{2!}=i^{2}dz_{1}\wedge d\bar{z}_{1}\wedge dz_{2}\wedge d\bar{z}_{2} \end{align*} $$

and since $\gamma _{\epsilon ,\delta }=i\partial \bar {\partial }F_{\epsilon }$ on $\widetilde {V}(\delta )_{\frak p}$ , we get

$$ \begin{align*}\left. \left( \frac{\det\gamma_{\epsilon,\delta}}{\det\gamma_{\epsilon(\rho)}}\right) \right|_{\widetilde{X}^{\theta}} = \left. \left( \frac{\gamma_{\epsilon,\delta}^{2}/2!}{\gamma_{\epsilon(\rho)}^{2}/2!}\right) \right|_{\widetilde{X}^{\theta}} = 1. \end{align*} $$

If $E_{i}\cong \mathbf {P}^{1}$ is a component of $\widetilde {X}^{\theta }$ , then $(E_{i},\gamma _{\epsilon ,\delta }|_{E_{i}})\cong (\mathbf {P}^{1},\epsilon \,\omega _{\mathrm {FS}})$ and $(E_{i},\gamma _{\epsilon (\rho )}|_{E_{i}})\cong (\mathbf {P}^{1},\epsilon (\rho )\omega _{\mathrm {FS}})$ . Hence

$$ \begin{align*}\left.\frac{\gamma_{\epsilon,\delta}}{\gamma_{\epsilon(\rho)}}\right|_{\widetilde{X}^{\theta}}=\frac{\epsilon}{\epsilon(\rho)}. \end{align*} $$

Altogether, we get

(7.3) $$ \begin{align} \begin{aligned} \,& \int_{\widetilde{X}^{\theta}} \widetilde{\mathrm{Td}}_{\theta}(T\widetilde{X};\gamma_{\epsilon,\delta},\gamma_{\epsilon(\rho)})^{(1,1)} = -\int_{\widetilde{X}^{\theta}} \frac{1}{12} \left. \log\left(\frac{\gamma_{\epsilon,\delta}}{\gamma_{\epsilon(\rho)}} \right|{}_{\widetilde{X}^{\theta}} \right) \{c_{1}(\widetilde{X}^{\theta},\gamma_{\epsilon,\delta})+c_{1}(\widetilde{X}^{\theta},\gamma_{\epsilon(\rho)})\} \\ &= -\frac{\log(\epsilon/\epsilon(\rho))}{6} \int_{\widetilde{X}^{\theta}}c_{1}(\widetilde{X}^{\theta}) = -\frac{\log(\epsilon/\epsilon(\rho))}{6}\chi(\widetilde{X}^{\theta}) = -\frac{k}{3}\,\log\frac{\epsilon}{\epsilon(\rho)}, \end{aligned} \end{align} $$

where we used the fact $\widetilde {X}^{\theta }=E_{1}\amalg \cdots \amalg E_{k}$ , $k=\#\mathrm {Sing}\,X$ , $E_{i}\cong \mathbf {P}^{1}$ . This, together with equation (7.1), yields that

$$ \begin{align*}\epsilon^{k/3}\tau_{\mathbf{Z}_{2}}(\widetilde{X},\gamma_{\epsilon,\delta})(\theta)\mathrm{Vol}(\widetilde{X},\gamma_{\epsilon,\delta}) = \epsilon(\rho)^{k/3}\mathrm{Vol}(\widetilde{X},\gamma_{\epsilon(\rho)})\, \tau_{\mathbf{Z}_{2}}(\widetilde{X},\gamma_{\epsilon(\rho)})(\theta) \end{align*} $$

is independent of $\epsilon ,\delta \in (0,1]$ with $0<\epsilon \delta ^{-2}\leq \epsilon (\rho )$ .

(Step 3) Let $\chi $ be another cut-off function to glue Eguchi-Hanson instanton to the initial Kähler form $\gamma _{0}$ on X (compare Sections 5.2.1 and 5.2.3). Then there exists $\epsilon (\chi )\in (0,1)$ such that the function $\phi ^{\prime }_{\epsilon ,\delta }(z):=\|z\|^{2}+\chi _{\delta }(\|z\|)E(z,\epsilon )$ on $V(\infty )\setminus \{0\}$ is a potential of a Kähler form on $T^{*}\mathbf {P}^{1}=\widetilde {V}(\infty )$ for any $\epsilon ,\delta \in (0,1]$ with $0<\epsilon \delta ^{-2}\leq \epsilon (\chi )$ . Let $\gamma ^{\prime }_{\epsilon ,\delta }$ be the families of Kähler forms on $\widetilde {X}$ constructed in the same way as in equation (4.12), using $\kappa ^{\prime }_{\epsilon ,\delta }:=i\partial \bar {\partial }\phi ^{\prime }_{\epsilon ,\delta }$ instead of $\kappa _{\epsilon ,\delta }$ . By Step 2, we get

$$ \begin{align*}\epsilon^{k/3}\tau_{\mathbf{Z}_{2}}(\widetilde{X},\gamma^{\prime}_{\epsilon,\delta})(\theta)\mathrm{Vol}(\widetilde{X},\gamma^{\prime}_{\epsilon,\delta}) = \epsilon(\chi)^{k/3}\mathrm{Vol}(\widetilde{X},\gamma^{\prime}_{\epsilon(\chi)})\, \tau_{\mathbf{Z}_{2}}(\widetilde{X},\gamma^{\prime}_{\epsilon(\chi)})(\theta) \end{align*} $$

for any $\epsilon ,\delta \in (0,1]$ with $0<\epsilon \delta ^{-2}\leq \epsilon (\chi )$ . To prove the independence of the limit $\lim _{\epsilon \to 0}\epsilon ^{k/3}\tau _{\mathbf {Z}_{2}}(\widetilde {X},\gamma _{\epsilon ,\delta })(\theta )\mathrm {Vol}(\widetilde {X},\gamma _{\epsilon ,\delta })$ from the choice of $\rho $ , we must prove

(7.4) $$ \begin{align} \epsilon(\rho)^{k/3}\mathrm{Vol}(\widetilde{X},\gamma_{\epsilon(\chi)})\, \tau_{\mathbf{Z}_{2}}(\widetilde{X},\gamma_{\epsilon(\chi)})(\theta) = \epsilon(\chi)^{k/3}\mathrm{Vol}(\widetilde{X},\gamma^{\prime}_{\epsilon(\chi)})\, \tau_{\mathbf{Z}_{2}}(\widetilde{X},\gamma^{\prime}_{\epsilon(\chi)})(\theta). \end{align} $$

We set $g_{0}=\gamma _{\epsilon (\rho )}$ and $g_{1}=\gamma ^{\prime }_{\epsilon (\chi )}$ in equation (7.2). By the same computation as in equation (7.3), we get

$$ \begin{align*}\int_{\widetilde{X}^{\theta}} \widetilde{\mathrm{Td}}_{\theta}(T\widetilde{X};\gamma_{\epsilon(\rho)},\gamma^{\prime}_{\epsilon(\chi)})^{(1,1)} = -\frac{k}{3}\log\frac{\epsilon(\chi)}{\epsilon(\rho)}. \end{align*} $$

This, together with equation (7.1), yields equation (7.4). This completes the proof.

Proof. Throughout the proof, we fix $ x \in \widetilde {X}_{3\delta }, \ \ z\in \widetilde {V}_{2\delta }, \ \ t>0.$ Since the Ricci curvature of $\gamma _{\epsilon , \delta }$ is bounded by Lemma 4.5, the Sobolev estimate, together with the Moser iteration technique combined with the finite propagation speed argument as in Cheeger-Gromov-Taylor [Reference Cheeger, Gromov and Taylor13], gives the uniform estimate

$$ \begin{align*}| K_{\epsilon}^q(t, x, z) | \leq Ae^{C(1+ \epsilon \delta^{-4})} \delta^{-4} e^{-\frac{\delta^2}{32t}}.\end{align*} $$

Indeed, the finite propagation speed technique gives us the $L^2$ estimate