2.3.1 Candelas–de la Ossa metrics on the local model.

Consider the space $\widehat{V}$ , which is the total space of the bundle

\begin{equation*} \widehat{V} = \mathcal{O}(-1) \oplus \mathcal{O}(-1) \rightarrow \mathbb{P}^1.\end{equation*}

On this space, we have two trivializations

\begin{equation*} (U, (\lambda, u,v)) \text{ and } (U', (\lambda', u', v')),\end{equation*}

with transition functions given by

\begin{equation*} \lambda' = \lambda^{-1}, \, u' = \lambda u, \, v' = \lambda v.\end{equation*}

Note that $\lambda$ is the coordinate on the base space $\mathbb{P}^1$ , while u,v are fiber coordinates.

It will be convenient to define the well-defined radius function $r: \widehat{V} \rightarrow [0,\infty)$ given by

\begin{equation*} r(\lambda, u, v) = (1 + |\lambda|^2)^{{1}/{3}} (|u|^2 + |v|^2)^{{1}/{3}}.\end{equation*}

Without the power of $\tfrac{1}{3}$ , this function measures the distance squared from a point to the zero section $E \simeq \mathbb{P}^1$ along the fibers using the Fubini–Study metric $\widehat{\omega}_{FS}$ . The power is introduced so that this radius function coincides with the radius of the Calabi–Yau cone metric on the blowdown, and this will be discussed later.

The space $(\widehat{V}, r)$ is also equipped with a family of scaling maps. Namely, for $R\gt0$ , we have the map $S_R: \widehat{V} \rightarrow \widehat{V}$ given by

\begin{equation*} S_R(\lambda,u,v) = (\lambda,R^{{3}/{2}}u, R^{{3}/{2}} v).\end{equation*}

The radius behaves as it should under the scaling, as we have

\begin{equation*} r \circ S_R = R \cdot r.\end{equation*}

In [Reference Candelas and de la OssaCdlO90], Candelas and de la Ossa look for a Kähler Ricci-flat metric $\widehat{\omega}_{\mathrm{co},1}$ on $\widehat{V}$ of the form

\begin{equation*} \widehat{\omega}_{\mathrm{co},1} = \sqrt{-1} \partial \overline{\partial} f(r^3) + 4 \widehat{\omega}_{\mathrm{FS}},\end{equation*}

where $\widehat{\omega}_{\mathrm{FS}}$ is the Fubini–Study metric on $\mathbb{P}^1$ , and $f(x)=f(r^3)$ is a smooth function. They show that imposing the condition of Kähler Ricci-flatness yields the following first-order ODE for f:

\begin{equation*}x(f'(x))^3+6(f'(x))^2=1.\end{equation*}

The solution admits an expansion [Reference Collins, Picard and YauCPY24] for $x \gg 1$ given in terms of $r=x^{{{1}/{3}}}$ by

\begin{equation*}f =c_0 r^2 + c_1 \log r + c_2 r^{-2} + c_3 r^{-4} + \cdots\end{equation*}

for constants $c_0, c_1, c_2, \ldots.$ Thus, after rescaling $\widehat{\omega}_{\mathrm{co},1}$ such that $c_0 = {\tfrac{1}{2}}$ , we have the following expansion for large radius $r \gg 1$ :

(4) \begin{equation} \widehat{\omega}_{\mathrm{co},1} - \tfrac{1}{2} \sqrt{-1} \partial \bar{\partial} r^2 = c_{-1} \widehat{\omega}_{\mathrm{FS}} + c_1 \sqrt{-1} \partial \bar{\partial} \log r + c_2 \sqrt{-1} \partial \overline{\partial} r^{-2} + c_3 \sqrt{-1} \partial \overline{\partial} r^{-4} + \cdots.\end{equation}

Next, we note that the space $\widehat{V}$ can be regarded as a small blowup of the space

\begin{equation*} V_0 = \Big\{ z \in \mathbb{C}^4 \mid \sum_i z_i^2 = 0 \Big\}.\end{equation*}

There is a (scaled) blowdown map $\pi: \widehat{V} \rightarrow V_0$ such that $\pi^{-1}(0)$ is the zero section $E=\{r=0\}$ , and the restriction

\begin{equation*} \pi: \hat{V} \setminus E \rightarrow V_0 \setminus \{ 0 \}\end{equation*}

is biholomorphic. The map $\pi$ has the explicit expression

(5) \begin{equation} \pi (\lambda, u, v) = \bigg( \frac{\lambda v + u}{\sqrt{2}}, -\sqrt{-1} \cdot \frac{\lambda v - u}{\sqrt{2}}, -\sqrt{-1} \cdot \frac{v + \lambda u}{\sqrt{2}}, -\frac{v - \lambda u}{\sqrt{2}} \bigg)\!.\end{equation}

Likewise, away from the singularity at the origin, we can see that

\begin{equation*} \pi^{-1} (z_1, z_2, z_3, z_4) = \bigg( \frac{z_1 - \sqrt{-1} z_2}{\sqrt{2}}, \frac{\sqrt{-1} (z_3 + \sqrt{-1} z_4)}{\sqrt{2}}, \sqrt{-1} \cdot \frac{z_3 - \sqrt{-1} z_4}{z_1 - \sqrt{-1} z_2} \bigg)\!.\end{equation*}

The function r on $\widehat{V}$ becomes $\| z \|^{{{2}/{3}}}$ on $V_0$ via the identification $\pi$ , in the sense that $r(\lambda, u, v) = \| \pi (\lambda, u, v)\|^{{{2}/{3}}}$ . For this reason, we will also denote

\begin{equation*}r = \| z \|^{{2}/{3}} , \quad r: V_0 \rightarrow [0,\infty).\end{equation*}

The space $V_0$ admits a Calabi–Yau cone metric

\begin{equation*} \omega_{\mathrm{co},0} = \tfrac{1}{2} \sqrt{-1} \partial \bar{\partial} r^2.\end{equation*}

We briefly discuss the cone-metric geometry on $(V_0, \omega_{\mathrm{co},0})$ . Observe that $V_0$ is closed under scalar multiplication and addition, so that $V_0$ is a cone. The metric $\omega_{\mathrm{co},0}$ is well known [Reference Candelas and de la OssaCdlO90] to be Kähler Ricci-flat and is a cone metric over the link

\begin{equation*} L = \{z \in V_0 \mid r(z) = 1\},\end{equation*}

and we may write

(6) \begin{equation} g_{\mathrm{co},0} = dr \otimes dr + r^2 \cdot g_L,\end{equation}

where $g_L$ is the pullback of a metric on L.

The link L is $S^3 \times S^2$ . To see this, we express the defining condition $\sum_i z_i^2 = 0$ of $V_0$ in real coordinates $x_i, y_i$ such that $z_i = x_i + \sqrt{-1}y_i$ for each $i \in \{1,2,3,4\}$ . We obtain

\[0 = \| x \|^2 - \| y \|^2 +2\sqrt{-1}\langle x, y \rangle, \]

where $x=(x_1,x_2,x_3,x_4) \in \mathbb{R}^4$ and $y=(y_1,y_2,y_3,y_4) \in \mathbb{R}^4$ . Expressed in these terms, $V_0$ is the set of all $(x,y) \in \mathbb{R}^8 \cong \mathbb{R}^4 \oplus \mathbb{R}^4$ on which $\| x \| = \| y \|$ and $\langle x, y \rangle = 0$ . Fixing $r^3=\| x \|^2 + \| y \|^2=2$ implies that $\| x \|=\| y \|=1$ . In particular, we have $x \in S^3 \subset \mathbb{R}^4$ . Then, for each such choice of x, the conditions $\langle x, y \rangle=0$ and $\| y \| = 1$ imply that y is in the unit 2-sphere centered at 0 in the tangent space $T_x S^3$ . Thus, the set $\{z \in V_0 \mid r(z)=2^{{{1}/{3}}}\}$ is diffeomorphic to the unit-sphere bundle contained in the tangent bundle $TS^3$ , which is trivial. Thus, $\{z \in V_0 \mid r(z)=2^{{{1}/{3}}}\} \cong S^3 \times S^2$ , and by rescaling we have $L \cong S^3 \times S^2$ as well.

Returning to $(\widehat{V}, \widehat{g}_{\mathrm{co},1})$ , we can rescale the area of the zero section $E \simeq \mathbb{P}^1$ to obtain a 1-parameter family of metrics. We will refer to this family of metrics as the Candelas–de la Ossa metrics $\widehat{g}_{\mathrm{co},a}$ on the small resolution. The metrics $\widehat{g}_{\mathrm{co},a}$ satisfy the following important properties.

1. (CO SR I) Normalization. For $a \gt 0$ , we have

   \begin{equation*} \widehat{g}_{\mathrm{co},a} = a^2 \cdot S_{a^{-1}}^* (\widehat{g}_{\mathrm{co},1}). \end{equation*}

2. (CO SR II) Asymptotically conical decay. There exists $C \gt 0$ independent of a such that for all $a \gt 0$ ,

   \begin{equation*} |(\pi^{-1})^* (\widehat{g}_{\mathrm{co},a}) - g_{\mathrm{co},0}|_{g_{\mathrm{co},0}} \leq C a^2 r^{-2}. \end{equation*}
   The asymptotic decay can be derived from (4) for $a=1$ . Pulling back the estimate when $a=1$ by $S^*_a$ gives the estimate for general a. The estimate implies that the Candelas–de la Ossa metrics $\widehat{g}_{\mathrm{co},a}$ converge uniformly to the cone metric $g_{\mathrm{co},0}$ on compact sets away from the zero section E.

For $R\gt 0$ , we will denote by $\widehat{T}(R)$ the ‘tube’

\[\widehat{T}(R) := \{r \leq R\} \subseteq \widehat{V} \]

around the zero section $E \simeq \mathbb{P}^1$ . On the cone $V_0$ , we will analogously refer to the ‘disc’

\[D_0(R) := \{r \leq R\} \subseteq V_0 \]

of radius R around 0.

Let $\widehat{d}_{\mathrm{co},a}$ and $d_{\mathrm{co},0}$ denote the induced distance functions on $\widehat{X}$ and $X_0$ by $\widehat{g}_{\mathrm{co},a}$ and $g_{\mathrm{co},0}$ , respectively. As warm-up for our proof of the convergence of metrics on the compact threefold, we will present a proof of convergence of the local models.

A proof of this fact also follows from the PDE estimates in [Reference Collins, Guo and TongCGT22] for general asymptotically conical Calabi–Yau metrics on small resolutions.

2.3.2 Balanced metrics on the small resolution.

Next, we state the properties that we will need from the Fu–Li–Yau metrics on the compact Calabi–Yau threefold $\widehat{X}$ . Let $\widehat{\omega}_\textrm{ CY}$ be a Kähler metric on the Kähler threefold $\widehat{X}$ . The Fu–Li–Yau gluing construction [Reference Fu, Li and YauFLY12] (see also [Reference Collins, Picard and YauCPY24] and [Reference ChuanChu12] for further details on $\widehat{\omega}_{\mathrm{FLY},a}$ for $a \neq 0$ ) produces a sequence of metrics $\widehat{\omega}_{\mathrm{FLY},a}$ for $0 \leq a \leq 1$ such that

\begin{equation*} d \widehat{\omega}_{\mathrm{FLY},a}^2 = 0, \quad [\widehat{\omega}_{\mathrm{FLY},a}^2]=[\widehat{\omega}_\textrm{CY}^2] \in H^4(\widehat{X},\mathbb{R}).\end{equation*}

For the purposes of this paper, we will mainly make use of the two following properties.

1. (FLY SR I) Local model. There exists $\delta\gt 0$ and $R\gt 1$ such that for all $0 \leq a \leq 1$ , we have

   \begin{equation*} \widehat{\omega}_{\mathrm{FLY},a}|_{ \{ r< \delta \}} = R \cdot \widehat{\omega}_{\mathrm{co},a}.\end{equation*}
   Here, the function $r: \widehat{X} \rightarrow [0,\infty)$ extends the local functions r defined on a neighborhood of the curves $E_i \subset \widehat{V}$ to the whole compact manifold $\widehat{X}$ such that the set $\{ r \lt \delta \}$ consists of small disjoint open neighborhoods containing the $(-1,-1)$ -curves $E_1, \dots, E_k$ .

2. (FLY SR II) Uniform convergence. For any compact set $K \subset \widehat{X} \setminus ( E_1 \cup \dots \cup E_k)$ , the sequence $\widehat{\omega}_{\mathrm{FLY},a}$ converges uniformly to $\widehat{\omega}_{\mathrm{FLY},0}$ as $a \rightarrow 0$ on K.

For each $E_i \simeq \mathbb{P}^1$ , these metrics satisfy

(7) \begin{equation} \int_{\mathbb{P}^1} \widehat{\omega}_{\mathrm{FLY},a} \rightarrow 0, \quad a \rightarrow 0.\end{equation}

The limiting metric $\widehat{\omega}_{\mathrm{FLY},0}$ is singular on the curves $E_1, \dots, E_k$ , and only defines a genuine metric on $\widehat{X} \backslash (E_1 \cup \dots \cup E_k)$ .

Let $\pi: \widehat{X} \rightarrow X_0$ be the contraction of the curves, and let $s_i = \pi(E_i)$ be the singular points of $X_0$ . We will write $(X_0)_\textrm{reg} = X_0 \backslash \{ s_1, \dots, s_k \}$ . Since $\widehat{X} \backslash (E_1 \cup \dots \cup E_k) \simeq (X_0)_\textrm{reg}$ , the limiting metric $\widehat{\omega}_{\mathrm{FLY},0}$ defines a Riemannian structure $( (X_0)_{\mathrm{reg}}, \, \omega_{\mathrm{FLY},0} )$ with conical singularities. This induces a distance function $d_{\mathrm{FLY},0}$ on $X_0$ .

This brings us to one of our main theorems.

2.3.3 Hermitian Yang–Mills metrics on the small resolution.

We review the relevant properties of the sequence of Hermitian Yang–Mills metrics from [Reference Collins, Picard and YauCPY24]. Recall that our initial threefold $\widehat{X}$ is Kähler Calabi–Yau and simply connected. By dimensional considerations, an application of the de Rham decomposition theorem [Reference YauYau93] implies that $(\widehat{X}, \widehat{\omega}_\textrm{CY})$ satisfies the stability condition

\begin{equation*} \frac{1}{\text{rank}\,F} \int_{\widehat{X}} c_1(F) \wedge \widehat{\omega}^2_\textrm{CY} \lt 0\end{equation*}

for all torsion-free coherent proper subsheaves $F \subseteq T^{1,0}\widehat{X}$ . The Fu–Li–Yau metric $\widehat{\omega}_{\mathrm{FLY},a}$ and the Calabi–Yau metric $\widehat{\omega}_\textrm{CY}$ have the same squared cohomology class, and so

\begin{equation*} \frac{1}{\text{rank}\,F} \int_{\widehat{X}} c_1(F) \wedge \widehat{\omega}^2_{\mathrm{FLY},a} \lt 0.\end{equation*}

It follows that $T^{1,0}\widehat{X}$ is stable with respect to each of the Fu–Li–Yau metrics. The Li–Yau generalization [Reference Li and YauLY87] of the Donaldson–Uhlenbeck–Yau theorem [Reference DonaldsonDon85, Reference Uhlenbeck and YauUY86] to Gauduchon metrics yields a family of metrics $\widehat{H}_a$ satisfying

(8) \begin{equation} F_{\widehat{H}_a} \wedge \widehat{\omega}^2_{\mathrm{FLY},a} = 0, \quad \int_{\hat{X}} \log \frac{\det \widehat{H}_a}{\det \widehat{g}_{\mathrm{FLY},a}} \, d\textrm {vol}_{g_{\mathrm{FLY},a}} = 0.\end{equation}

This sequence $\widehat{H}_a$ satisfies the following estimates.

Proposition 1 (Collins, Picard and Yau [Reference Collins, Picard and YauCPY24]). There exist constants $C, C_k \gt 0$ such that the Hermitian Yang–Mills metrics $\widehat{H}_a$ satisfy

\begin{align*} C^{-1} \cdot \widehat{g}_{\mathrm{FLY},a} &\leq \widehat{H}_a \leq C \cdot \widehat{g}_{\mathrm{FLY},a},\\ |\nabla^k \widehat{H}_a|_{\widehat{g}_{\mathrm{FLY},a}} &\leq C_k r^{-k}.\end{align*}

The metric $H_0$ on $(X_0)_{\mathrm{reg}}$ can be constructed as the limit of these metrics $\widehat{H}_a$ . This was done in [Reference Collins, Picard and YauCPY24] by taking a subsequence of $\{ \widehat{H}_a \}$ . In Appendix A, we will show that these estimates imply that the full sequence converges on compact sets. Therefore, there exists a Hermitian Yang–Mills metric $H_0$ over the singular space $X_0$ such that

\begin{equation*} F_{H_0} \wedge \omega^2_{\mathrm{FLY,0}} = 0, \quad C^{-1} \cdot g_{\mathrm{FLY},0} \leq H_0 \leq C \cdot g_{\mathrm{FLY},0}\end{equation*}

and for any compact set $K \subset \widehat{X} \setminus (E_1 \cup \dots \cup E_k)$ , the sequence $\widehat{H}_a$ converges uniformly to $H_0$ on K as $a \rightarrow 0$ . Going beyond compact subsets, we will prove the following.

2.4.1 Candelas–de la Ossa metrics on the local model.

Next, for $t \neq 0$ , we consider the smooth submanifolds $V_t \subset \mathbb{C}^4$ defined by

\begin{equation*} V_t = \Big\{ z \in \mathbb{C}^4 \mid \sum_i z_i^2 = t \Big\}.\end{equation*}

We have the usual norm $\|z\|^2$ on $V_t$ induced from $\mathbb{C}^4$ , given by

\begin{equation*} \|z\|^2 = \sum_{i=1}^4 |z_i|^2.\end{equation*}

Candelas and de la Ossa also constructed metrics $\omega_{\mathrm{co},t}$ on the smoothings $V_t$ [Reference Candelas and de la OssaCdlO90]. The metrics are obtained by looking for potentials of the form

\begin{equation*} \omega_{\mathrm{co},t} = \sqrt{-1} \partial \overline{\partial} \varphi_t, \quad \varphi_t = f_t(\| z \|^2)\end{equation*}

and imposing the Ricci-flat condition, which reduces to solving a differential equation for $f_t$ . These metrics $\omega_{\mathrm{co},t}$ are asymptotic to the cone geometry $(V_0,\omega_{\mathrm{co},0})$ , and we will make this more precise below.

As we did in the previous section, we define a radius function $r: V_t \rightarrow (0,\infty)$ by

\begin{equation*} r(z) = \| z \|^{{2}/{3}}.\end{equation*}

Note that the condition $\sum_{i=1}^4 z_i^2 = t$ implies that $r(z) \geq |t|^{{{1}/{3}}}$ for all $z \in V_t$ .

When $R \gt 0$ , we may also define scaling maps $S_R \colon \mathbb{C}^4 \rightarrow \mathbb{C}^4$ by

\begin{equation*} S_R(z) = R^{{3}/{2}} \cdot z.\end{equation*}

The scaling map $S_R$ sends $V_\rho$ to $V_{R^3 \cdot \rho}$ and satisfies

\begin{equation*} r \circ S_R = R \cdot r \quad \text{ and } \quad S_R^* (\omega_{\mathrm{co},0}) = R^2 \cdot \omega_{\mathrm{co},0}.\end{equation*}

Unlike the case of the small resolutions, the metrics $\omega_{\mathrm{co},t}$ and $\omega_{\mathrm{co},0}$ all lie on different spaces. In order to compare them (and obtain an analog of Property (CO SR II)), we use the map $\Phi \colon \mathbb{C}^4 \setminus \{0\} \rightarrow \mathbb{C}^4$ given by

\begin{equation*} \Phi(z) = z + \frac{\overline{z}}{2 \|z\|^2}.\end{equation*}

Routine computations show that $\Phi$ maps $V_0 \setminus \{0\}$ into $V_1$ . This map is not injective in general, but is a diffeomorphism when restricted to the set $\{z \in V_0 \mid r(z) \gt ({\tfrac{1}{2}})^{{{1}/{3}}} \}$ with image $\{z \in V_1 \mid r(z) \gt 1 \}$ . In the sequel, $\Phi$ will refer to this restricted map and we will use results involving the map $\Phi$ as required. More details regarding them can be found in Appendix B.

By composing $\Phi$ with $S_R$ for appropriate choices of R, we get a map from $V_0 \setminus \{0\}$ to $V_t$ . More precisely, let

(9) \begin{equation} \Phi_t = S_{t^{{1}/{3}}} \circ \Phi \circ S_{t^{-{1}/{3}}}.\end{equation}

Here we used the scaling maps to scale by a root of a non-zero complex number. Note that, although $S_{t^{{{1}/{3}}}}$ depends on a choice of branch, the map $\Phi_t$ does not. We can compute that for $z \in V_0$ ,

\begin{equation*} r(\Phi_t(z)) = \Big( (r(z))^3 + \frac{|t|^2}{4 (r(z))^3} \Big)^{{{1}/{3}}}.\end{equation*}

In view of this identity, it will be convenient later on (especially in §4.4) to use the notation

\begin{equation*} \beta_{t,\rho} = \Big( \rho^3 + \frac{|t|^2}{4\rho^3} \Big)^{{{1}/{3}}}.\end{equation*}

We also can check that $\Phi_t$ is a diffeomorphism from $\{z \in V_0 \mid r(z) \gt ({|t|}/{2})^{{{1}/{3}}} \}$ to $\{z \in V_t \mid r(z) \gt |t|^{{{1}/{3}}} \}$ . We use $\Phi_t$ to refer to the restricted map.

Using the maps $S_R$ and $\Phi_t$ , we have the analogous properties of the Candelas–de la Ossa metrics $g_{\mathrm{co},t}$ on the smoothings.

1. (CO SM I) Normalization. For $t \neq 0$ ,

   \begin{equation*} g_{\mathrm{co},t} = |t|^{{2}/{3}} \cdot S_{t^{-{1}/{3}}}^* (g_{\mathrm{co},1}). \end{equation*}
   The metrics $g_{\mathrm{co},t}$ do not depend on the branch choice of $t^{-{{1}/{3}}}$ .

2. (CO SM II) Asymptotically conical decay. There exists $C \gt 0$ independent of t such that for all $t \neq 0$ ,

   \begin{equation*} |(\Phi_t)^* (g_{\mathrm{co},t}) - g_{\mathrm{co},0}|_{g_{\mathrm{co},0}} \leq C |t| r^{-3}. \end{equation*}
   A consequence of this is that the metrics $g_{\mathrm{co},t}$ approach $g_{\mathrm{co},0}$ on compact sets away from the cone singularity as $t \rightarrow 0$ .

The proof of the asymptotically conical decay estimate can be found in e.g. [Reference Conlon and HeinCH13], where the estimate is given on $(V_1,g_{\mathrm{co},1})$ :

(10) \begin{equation} | \Phi^* g_{\mathrm{co},1} - g_{\mathrm{co},0}|_{g_{\mathrm{co},0}} \leq C r^{-3},\end{equation}

and the estimate for $(V_t, g_{\mathrm{co},t})$ follows by pulling back via $S_{t^{-{{1}/{3}}}}$ .

Let

\begin{equation*} D_t(R) := \{r \leq R \} \subseteq V_t\end{equation*}

be ‘discs’ of radius R in $V_t$ .

As warm-up to our result on continuity of the global non-Kähler geometry, we will present the following well-known convergence of the local models.

2.4.2 Balanced metrics on the smoothings.

We return to the global setting, where we have a holomorphic family $\mu: \mathcal{X} \rightarrow \Delta$ with smooth fibers $X_t = \mu^{-1}(t)$ for $t \neq 0$ and central fiber $X_0$ with singularities $\{s_1, \dots , s_k \}$ , which are locally of the form $0 \in V_0$ . If $X_0$ comes from holomorphic contraction $\pi: \widehat{X} \rightarrow X_0$ of $(-1,-1)$ -curves, the compact complex manifolds $X_t$ may not support any Kähler metric. Fu, Li and Yau [Reference Fu, Li and YauFLY12] prove that $X_t$ admits balanced metrics.

In order to define metrics on the smoothings $X_t$ , we first need to extend the local maps r and $\Phi_t$ from the previous section to global objects.

For this, we note that there are disjoint open sets $\mathcal{U}_i \subset \mathcal{X}$ containing each singular point $s_i$ such that $\mathcal{U}_i$ is identified with

\begin{equation*}0 \in \mathcal{U} \subset \Big\{ (z,t) \in \mathbb{C}^4 \times \mathbb{C} \mid \sum_i z_i^2 = t \Big\}.\end{equation*}

We can then extend the local functions $r = \| z \|^{{{2}/{3}}}$ on $\mathbb{C}^4$ to a global function $r: \mathcal{X} \rightarrow [0,\infty)$ with $r^{-1}(0) = \{ s_1, \dots, s_k \}$ .

Next, we can extend the local maps $\Phi_t$ to global diffeomorphisms

\begin{equation*} \Phi_t: X_0 \cap \Big\{ r(z) \gt \Big(\frac{|t|}{2} \Big)^{{1}/{3}} \Big\} \rightarrow X_t \cap \{ r(z) \gt |t|^{{1}/{3}} \}\end{equation*}

such that $\Phi_t$ is the model smoothing (9) on the local sets $\mathcal{U}_i$ . This can be done by taking a horizontal lift $\xi$ of the vector field ${\partial}/{\partial t}$ on $\Delta$ , which agrees with the vector field generating the model smoothing. Then, flowing by the lifted vector field $\xi$ on $\mathcal{X}$ gives $\Phi_t$ .

The Fu–Li–Yau construction [Reference Fu, Li and YauFLY12] leads to a sequence $\omega_{\mathrm{FLY},t}$ of Hermitian metrics on $X_t$ solving

\begin{equation*} d \omega_{\mathrm{FLY},t}^2 = 0.\end{equation*}

These are obtained by: (1) a pullback and gluing construction, followed by (2) a perturbation step to ensure the balanced condition. The metric in step 1 is denoted $g_t$ and the metric in step 2 is denoted $g_{\mathrm{FLY},t}$ .

${\unicode{x2013}}$ Step 1. The expression for $\omega_t$ from [Reference Fu, Li and YauFLY12] is

(11) \begin{equation} \begin{aligned} \omega_t^2 &= pr_t^{2,2} [ (\Phi_t^{-1})^* (\omega_{\mathrm{FLY},0}^2 - \sqrt{-1} \partial \overline{\partial} (\rho_0 \cdot f_0 (\|z\|^2) \cdot \sqrt{-1} \partial \overline{\partial} f_0 (\|z\|^2))) \\ &\quad \quad + \sqrt{-1} \partial \overline{\partial}( \rho_t \cdot f_t (\|z\|^2) \cdot \sqrt{-1} \partial \overline{\partial} f_t (\|z\|^2))], \end{aligned} \end{equation}

where $pr_t^{2,2}$ denotes the projection onto the (2,2) component with respect to the complex structure $J_t$ on $X_t$ and $\rho_0$ and $\rho_t$ are smooth cutoff functions. In the above, $f_t$ is the function

\begin{equation*} f_t(x) = \Big( \frac{|t|^2}{2} \Big)^{{1}/{3}} \int_0^{\cosh^{-1} (x/|t|)} (\sinh (2y) - 2y)^{{1}/{3}} dy. \end{equation*}

These functions are from the Calabi–Yau local model $\omega_{\mathrm{co},t} = \sqrt{-1} \partial \bar{\partial} f_t$ , so that in a neighborhood $\{r \lt \delta \}$ where the cutoff functions $\rho \equiv 1$ , there holds $\omega^2_t = \omega^2_{\mathrm{co},t}$ . We take a square root of (11) to obtain the metric $\omega_t$ .

${\unicode{x2013}}$ Step 2. The Fu–Li–Yau metrics correct $\omega_t$ by

(12) \begin{equation} \omega_{\mathrm{FLY},t}^2 = \omega_t^2 + \partial \bar{\partial}^\dagger \partial^\dagger \gamma_t - \bar{\partial} \partial^\dagger \bar{\partial}^\dagger \bar{\gamma_t}\end{equation}

where $\gamma_t \in \Lambda^{2,3}(X_t)$ solves

\[E_t(\gamma_t) = \bar{\partial} \omega_t^2, \quad d \gamma_t = 0 \]

and $E_t$ is the Kodaira–Spencer operator [Reference Kodaira and SpencerKS60], which is a fourth-order elliptic operator that acts on (2,3)-forms by

\[E_t = \partial \bar{\partial} \bar{\partial}^\dagger \partial^\dagger + \partial^\dagger \bar{\partial} \bar{\partial}^\dagger \partial + \partial^\dagger \partial . \]

The adjoints here are with respect to $\omega_t$ . The construction is such that $d \omega_{\mathrm{FLY},t}^2 = 0$ , and the main part of the argument in [Reference Fu, Li and YauFLY12] is to prove that $\omega_{\mathrm{FLY},t} \gt 0$ for small enough t.

We will need the following two properties of the Fu–Li–Yau metrics. These properties can be extracted from the estimates in [Reference Fu, Li and YauFLY12], and we refer to [Reference Collins, Picard and YauCPY24] for further discussion.

1. (FLY SM I) Local model. Near each singular point $s_i \in \mathcal{X}$ , there exist constants $c,C,\epsilon,\delta\gt 0$ and such that for all $t \in \Delta_\epsilon$ , we have

   (13) \begin{equation}\sup_{ \{ r \leq \delta \}} | g_{\mathrm{FLY},t} - c \cdot g_{\mathrm{co},t}|_{g_{\mathrm{co},t}} \leq C |t|^{{2}/{3}}.\end{equation}
   Here, $\Delta_\epsilon = \{ t \in \mathbb{C} : |t|< \epsilon \}$ .

2. (FLY SM II) Uniform convergence. For any compact set $K \subset (X_0)_{\mathrm{reg}}$ , the sequence $\Phi_t^* g_{\mathrm{FLY},t}$ converges uniformly to $g_{\mathrm{FLY},0}$ as $t \rightarrow 0$ on K.

Our main theorem on the smoothings takes the following form.

2.4.3 Hermitian Yang–Mills metrics on the smoothing.

In order to get approximate Hermitian Yang–Mills solutions on the smoothings, Collins, Picard and Yau glued the pullback of the metric $H_0$ to the Candelas–de la Ossa metrics [Reference Collins, Picard and YauCPY24]. These approximate metrics were perturbed to obtain true solutions $H_t$ to the Hermitian Yang–Mills equations. The resulting metrics $H_t$ on $X_t$ solve

\begin{equation*} F_{H_t} \wedge \omega^2_{\mathrm{FLY},t} =0, \quad \int_{X_t} \log \frac{\det H_t}{\det g_{\mathrm{FLY},t}} \, d \textrm{vol}_{g_{\mathrm{FLY},t}} = 0.\end{equation*}

The construction of [Reference Collins, Picard and YauCPY24] is such that

\begin{equation*}H_t = H_{t,\textrm{ref}} e^{u}, \quad | u |_{H_{t, \textrm{ref}}} + r |\nabla u|_{H_{t, \textrm{ref}}} \leq C |t|^{(1/3) |\beta|}\end{equation*}

for $|\beta| \in (0,1)$ , and

\begin{equation*} H_{t,\textrm{ref}} = \chi g_{\mathrm{co},t} + (1-\chi) [(\Phi_t^{-1})^* H_0]^{1,1}\end{equation*}

and $\chi(z) = \zeta( |t|^{-\alpha} \| z \|^2 )$ for $0<\alpha<1$ and $\zeta: [0,\infty) \rightarrow [0,1]$ with $\zeta \equiv 1$ on [0,1] and $\zeta \equiv 0$ on $[2,\infty)$ .

The metrics $H_t$ are uniformly equivalent to $g_{\mathrm{FLY},t}$ , so that

(14) \begin{equation} C^{-1} \cdot g_{\mathrm{FLY},t} \leq H_t \leq C \cdot g_{\mathrm{FLY},t}.\end{equation}

Furthermore, for any compact set $K \subset (X_0)_{\mathrm{reg}}$ , the sequence $\Phi_t^* H_t$ converges uniformly to $H_0$ as $t \rightarrow 0$ on K. We will show the Gromov–Hausdorff convergence of the Yang–Mills metrics.

4.3.1 Tubular bounds.

Recall the asymptotically conical decay property (CO SR II). We may fix a constant K such that

\begin{equation*} |(\pi^{-1})^* (\widehat{g}_{\mathrm{co},a}) - g_{\mathrm{co},0}|_{g_{\mathrm{co},0}} \leq \tfrac{1}{2}\end{equation*}

when $r \gt aK$ . We start with uniform bounds on the spaces $(\widehat{T}(aK), \widehat{g}_{\mathrm{co},a})$ . These will be obtained using the scaling property (CO SR I) and the compactness of the set $(\widehat{T}(K), \widehat{g}_{\mathrm{co},1})$ .

To estimate the diameter, we consider a curve $\gamma \colon [0,1] \rightarrow \widehat{T}(aK)$ . The length of this curve with respect to the metric $\widehat{g}_{\mathrm{co},a}$ is given by

\begin{equation*}\begin{aligned} \widehat{L}_{\mathrm{co},a} (\gamma) &= \int_0^1 \sqrt{\widehat{g}_{\mathrm{co},a} ( \dot{\gamma}(s), \dot{\gamma}(s))} \, ds \\ &= \int_0^1 \sqrt{ a^2 \cdot S^*_{a^{-1}} (\widehat{g}_1)( \dot{\gamma}(s), \dot{\gamma}(s) )} \, ds \\ &= a \cdot \int_0^1 \sqrt{ \widehat{g}_{\mathrm{co},1} ( (S_{a^{-1}})_* \dot{\gamma}(s), (S_{a^{-1}})_* \dot{\gamma}(s) )} \, ds \\ &= a \cdot \widehat{L}_{\mathrm{co},1} ( S_{a^{-1}} \circ \gamma ),\end{aligned}\end{equation*}

where we have used (CO SR I). Since there is a one-to-one correspondence between curves in $\widehat{T}(aK)$ and curves in $\widehat{T}(K)$ given by composition with $S_{a^{-1}}$ , it follows that

(36) \begin{equation} \widehat{\text{diam}}_{\mathrm{co},a} (\widehat{T}(aK)) = a \cdot \widehat{\text{diam}}_{\mathrm{co},1} (\widehat{T}(K)).\end{equation}

To obtain a volume bound, we note that

\begin{equation*} \widehat{\text{{Vol}}}_{\mathrm{co},a} (\widehat{T}(aK)) = \int_{\widehat{T}(aK)} \widehat{\omega}_{\mathrm{co},a}^3 = \int_{\widehat{T}(aK)} a^6 \cdot S^*_{a^{-1}} (\widehat{\omega}_{\mathrm{co},1}^3).\end{equation*}

Using the change of variables formula, this becomes

\begin{equation*} \widehat{\text{{Vol}}}_{\mathrm{co},a} (\widehat{T}(aK)) = a^6 \cdot \int_{S_{a^{-1}}(\widehat{T}(aK))} \widehat{\omega}_{\mathrm{co},1}^3 = a^6 \cdot \int_{\widehat{T}(K)} \widehat{\omega}_{\mathrm{co},1}^3 .\end{equation*}

Therefore,

(37) \begin{equation} \widehat{\text{{Vol}}}_{\mathrm{co},a} (\widehat{T}(aK)) = a^6 \cdot \widehat{\text{{Vol}}}_{\mathrm{co},1} (\widehat{T}(K)).\end{equation}

4.3.2 Annular bounds.

Let $\delta \gt 0$ . We will obtain diameter and volume bounds on the annular region $\widehat{T}(\delta) \backslash \widehat{T}(aK)$ for $0 \lt a \leq {\delta}/{K}$ . These are derived using the asymptotically conical decay property (CO SR II).

Fix a point $p = (\lambda, u_0, v_0) \in \widehat{T}(\delta) \backslash \widehat{T}(aK)$ , and denote $\rho = r(p)$ . Then $\rho \in (aK,\delta]$ .

Consider the curve $\widehat{\gamma}: [({a}/{\rho})K, 1] \rightarrow \widehat{T}(\delta)$ given by

(38) \begin{equation} \widehat{\gamma}(s) = (\lambda_0, s^{{3}/{2}} u_0, s^{{3}/{2}} v_0).\end{equation}

This path begins in $\widehat{T}(aK)$ and moves along the fiber over $\lambda_0$ to arrive at $p=\widehat{\gamma}(1)$ .

Using the blowdown map $\pi: \widehat{V} \rightarrow V_0$ (5), it can be directly checked that this curve is sent to the curve $\gamma = \pi \circ \widehat{\gamma}$ in $V_0$ given by

\begin{equation*} \gamma(s) = s^{{3}/{2}} \cdot \pi(\lambda_0, u_0, v_0).\end{equation*}

It follows that

\begin{equation*} r (\gamma (s)) = s \cdot \rho.\end{equation*}

Proof. The cone metric can be written as $g_{\mathrm{co},0} = dr^2 + r^2 \cdot \textrm{pr}_1^* g_L$ , where $g_L$ is a metric on the link $L = \{ r = 1\}$ and $\textrm{pr}_1: V_0 \rightarrow L$ is the projection to the link $\textrm{pr}_1(z) = {z}/{\| z \|}$ . We can then compute

\begin{equation*}dr (\dot{\gamma}) = \frac{d}{ds} (r \circ \gamma) = \rho\end{equation*}

and

\begin{equation*} (\textrm{pr}_1)_* \dot{\gamma} = \frac{d}{ds} (\textrm{pr}_1 \circ \gamma) = 0.\end{equation*}

Therefore,

\begin{equation*} g_{\mathrm{co},0}(\dot{\gamma},\dot{\gamma}) = \rho^2.\end{equation*}

This gives the speed of $\gamma$ , and integration gives the length.

We now compare the length of the curve $(\widehat{\gamma}, \widehat{g}_{\mathrm{co},a})$ to the length of the curve $(\gamma, g_{\mathrm{co},0})$ :

\begin{equation*}\begin{aligned} |\widehat{L}_{\mathrm{co},a}(\widehat{\gamma}) - L_{\mathrm{co},0} (\gamma)| &= \Bigg| \int_{({a}/{\rho})K}^1 |\dot{\widehat{\gamma}}|_{\widehat{g}_{\mathrm{co},a}} - |\dot{\gamma}|_{g_{\mathrm{co},0}} \, ds \Bigg| \\ &\leq \int_{({a}/{\rho)}K}^1 \bigg| |\dot{\widehat{\gamma}}|_{\widehat{g}_{\mathrm{co},a}} - |\dot{\gamma}|_{g_{\mathrm{co},0}} \bigg| \, ds \\ &= \int_{({a}/{\rho})K}^1 \bigg| |\dot{\gamma}|_{(\pi^{-1})* (\widehat{g}_{\mathrm{co},a})} - |\dot{\gamma}|_{g_{\mathrm{co},0}} \bigg| \, ds \\ &= \int_{({a}/{\rho})K}^1 \Bigg| \frac{|\dot{\gamma}|^2_{(\pi^{-1})* (\widehat{g}_{\mathrm{co},a})} - |\dot{\gamma}|^2_{g_{\mathrm{co},0}}}{|\dot{\gamma}|_{(\pi^{-1})* (\widehat{g}_{\mathrm{co},a})} + |\dot{\gamma}|_{g_{\mathrm{co},0}}} \Bigg| \, ds.\end{aligned}\end{equation*}

We then obtain the estimate

\begin{equation*} \begin{aligned}|\widehat{L}_{\mathrm{co},a}(\widehat{\gamma}) - L_{\mathrm{co},0} (\gamma)| &\leq \int_{({a}/{\rho})K}^1 \frac{| (\pi^{-1})^* \widehat{g}_{\mathrm{co},a} - g_{\mathrm{co},0}|_{g_{\mathrm{co},0}} \, |\dot{\gamma}|_{g_{\mathrm{co},0}}^2}{|\dot{\gamma}|_{(\pi^{-1})* (\widehat{g}_{\mathrm{co},a})} + |\dot{\gamma}|_{g_{\mathrm{co},0}}} \, ds \\&\leq \int_{({a}/{\rho})K}^1 | (\pi^{-1})^* \widehat{g}_{\mathrm{co},a} - g_{\mathrm{co},0}|_{g_{\mathrm{co},0}} \, |\dot{\gamma}|_{g_{\mathrm{co},0}} \, ds. \end{aligned}\end{equation*}

We now use $|\dot{\gamma}|_{g_{\mathrm{co},0}}=\rho$ , $r(\gamma(s)) = s \cdot \rho$ and (CO SR II) to obtain

\begin{equation*}|\widehat{L}_{\mathrm{co},a}(\widehat{\gamma}) - L_{\mathrm{co},0} (\gamma)|\leq \int_{({a}/{\rho})K}^1 \frac{C a^2 \rho}{r^2} \, ds = \frac{Ca^2}{\rho} \cdot \int_{({a}/{\rho})K}^1 \frac{1}{s^2} \, ds = Ca \cdot \Big( \frac{1}{K} - \frac{a}{\rho} \Big).\end{equation*}

Therefore,

\begin{equation*} \widehat{L}_{\mathrm{co},a}(\widehat{\gamma}) \leq |\widehat{L}_{\mathrm{co},a}(\widehat{\gamma}) - L_{\mathrm{co},0} (\gamma)| + |L_{\mathrm{co},0}(\gamma)| \leq Ca \cdot \Big( \frac{1}{K} - \frac{a}{\rho} \Big) + (\rho - aK).\end{equation*}

For fixed $0 \lt a \leq {\delta}/{K}$ , this is maximized when $\rho = \delta$ , giving

\begin{equation*} \widehat{L}_{\mathrm{co},a}(\widehat{\gamma}) \leq C \cdot \Big(\frac{a}{K\delta} + 1\Big) \cdot (\delta - aK) \leq C \cdot (\delta - aK).\end{equation*}

As such, we get

\begin{equation*} \widehat{d}_{\mathrm{co},a} (p, \widehat{T}(aK)) \leq C \cdot (\delta - aK).\end{equation*}

In tandem with our diameter bound for $\widehat{T}(aK)$ (36), we get

\begin{equation*} \widehat{\text{diam}}_{\mathrm{co},a} (\widehat{T}(\delta)) \leq a \cdot \widehat{\text{diam}}_{\mathrm{co},1} (\widehat{T}(K)) + 2C \cdot (\delta - aK),\end{equation*}

which for fixed $\delta$ and K is uniformly bounded for $0 \lt a \leq {\delta}/{K}$ .

In the appendix, we will need a volume estimate on tubes around the exceptional divisor. This volume estimate follows from the diameter estimate and the Bishop–Gromov comparison theorem. Indeed, for $0 \lt a \leq {\delta}/{K}$ , the diameter estimate tells us that

\[\widehat{\text{diam}}_{\mathrm{co},a} (\widehat{T}(\delta)) \leq C \delta. \]

Thus, the tube $\widehat{T}(\delta)$ is contained in the ball $\widehat{B}(p,C\delta)$ for any point $p \in \widehat{T}(\delta)$ . Therefore,

\begin{align*} \widehat{\text{Vol}}_{\mathrm{co},a}(\widehat{T}(\delta)) &\leq \widehat{\text{Vol}} \, \widehat{B}(p,C\delta) \\ &\leq \text{Vol}_{\mathrm{Euc}} \,B(0,C\delta),\end{align*}

where $\text{Vol}_{\mathrm{Euc}} \, B(0,C\delta)$ denotes the Euclidean volume of the ball of radius $C\delta$ in $\mathbb{R}^6$ . This second inequality is by Bishop–Gromov, together with the fact that the metrics $\widehat{g}_{\mathrm{co},a}$ are Ricci-flat. Thus, $ \widehat{\text{Vol}}_{\mathrm{co},a}(\widehat{T}(\delta)) \leq C \delta^6$ . We record these diameter and volume bounds for future reference. The diameter bound is used in the next section, and the volume bound is used in the appendix.

Lemma 7. For $\delta \gt 0$ , we have

(39) \begin{equation} \widehat{\text{diam}}_{\mathrm{co},a} (\widehat{T}(\delta)) \leq C \delta \end{equation}

and

(40) \begin{equation} \widehat{\textrm {Vol}}_{\mathrm{co},a}(\hat{T}(\delta)) \leq C \delta^6 \end{equation}

for any $0 \lt a \leq {\delta}/{K}$ .

4.3.3 Applying the main lemma.

Our diameter estimates will enable us to prove the following useful lemma akin to that of Song and Weinkove [Reference Song and WeinkoveSW13].

Next, we consider the second condition, and note $\pi^{-1}(D_0(\delta)) = \widehat{T}(\delta)$ . Using (39), we see that the result follows once we choose $\delta$ small enough such that $\delta \lt C^{-1} \epsilon$ , $\delta \lt {\epsilon}/{2}$ and $a_0 = {\delta}/{K}$ .

We can now apply Lemma 5 to prove convergence of the three classes of metrics on the small resolution.

• – Theorem 2.7. Convergence of the local models $(\widehat{T}(1),\widehat{d}_{\mathrm{co},a}) \rightarrow (D_0(1), d_{\mathrm{co},0})$ .

  In this case, we have only one ODP singularity s. By the diameter estimate of $g_{\mathrm{co},a}$ (Lemma 8), we see that for each $\epsilon \gt 0$ , we can pick the set $\overline{G} = D_0(\delta)$ for an appropriately small $\delta \gt 0$ such that Lemma 5 applies.

• – Theorem 2.8. Convergence of the global balanced metrics $(\widehat{X}, \widehat{d}_{\mathrm{FLY},a}) \rightarrow (X_0, d_{\mathrm{FLY},0})$ .

  Here we use the fact that the Fu–Li–Yau metrics are, up to scaling, just the Candelas–de la Ossa metrics in a compact set around the $(-1,-1)$ -curves $E_i$ and the ODP singularities $s_i$ . For $\epsilon \gt 0$ , we can pick $\overline{G_i} = D_0(\delta_i)$ for appropriately small $\delta_i$ around each singular point $s_i$ . Coupling this with the smooth convergence of the Fu–Li–Yau metrics on compact sets away from the $(-1,-1)$ -curves, we may apply Lemma 5.

• – Theorem 2.12. Convergence of the global Hermitian Yang–Mills metrics $(\widehat{X},\widehat{d}_{\widehat{H}_a}) \rightarrow (X_0,d_{H_0})$ .

  By Proposition 1, we have the estimate

  \begin{equation*} C^{-1} \cdot g_{\mathrm{co},a} \leq \widehat{H}_a \leq C \cdot g_{\mathrm{co},a} \end{equation*}
  on the local sets $D_0(\delta_i)$ around each singularity $s_i$ , where the Fu–Li–Yau metrics are a scaling of the Candelas–de la Ossa metrics. Lemma 8 implies that, for $\epsilon \gt 0$ , there exist $\delta_i \gt 0$ and $a_0 \gt 0$ such that for all $0 \lt a \lt a_0$ ,
  \begin{equation*} \mathrm{diam}_{H_0}(D_0(\delta_i)) \lt \epsilon, \quad \widehat{\text{diam}}_{\widehat{H}_a}(\pi^{-1} (D_0(\delta_i))) \lt \epsilon. \end{equation*}
  We may therefore apply Lemma 5.

4.4.1 Bounds on the disc.

We start by estimating the geometry of the disc $D_t(\beta_{t, |t|^{{{1}/{3}}}K})$ . For this, we note that

\begin{equation*} S_{t^{-{1}/{3}}}: (D_t(\beta_{t,|t|^{{1}/{3}}K}),g_{\mathrm{co},t}) \rightarrow (D_1(\beta_{1,K}), |t|^{{2}/{3}} g_{\mathrm{co},1})\end{equation*}

is an isometry. This is due to the scaling property $g_{\mathrm{co},t} = |t|^{2/3} S^*_{t^{-1/3}}(g_{\mathrm{co},1})$ . It follows that

(41) \begin{equation} \text{diam}_{\mathrm{co},t} (D_t(\beta_{t,|t|^{{1}/{3}}K})) = |t|^{{1}/{3}} \cdot \text{diam}_{\mathrm{co},1} (D_1(\beta_{1,K})).\end{equation}

4.4.2 Annular bounds.

Let $\delta \gt 0$ . We now compute diameter bounds on the ‘annular’ region

$D_t(\beta_{t,\delta}) \backslash D_t(\beta_{t,|t|^{{{1}/{3}}}K})$ when $0 \lt |t| \leq ({\delta}/{K})^3$ . As before, this relies on (CO SM II).

Let $q \in D_t(\beta_{t,\delta}) \backslash D_t(\beta_{t, |t|^{{{1}/{3}}}K})$ be an arbitrary point in the annular region. We will construct a curve $\tilde{\gamma}$ from $D_t(\beta_{t, |t|^{{{1}/{3}}}K})$ to q and estimate its length $L_{\mathrm{co},t}(\widetilde{\gamma})$ . To do this, we will bring the setup back to the cone $V_0$ and use a radial ray.

Since $\Phi_t$ is a diffeomorphism on the annular region, we can write $q = \Phi_t(p)$ for $p \in V_0$ . We note that $\beta$ is defined such that $|t|^{{{1}/{3}}}K \lt r(p) \leq \delta$ and we define $\rho\gt 0$ by $r(p) = \rho$ . Hence, $|t|^{1/3} K \lt \rho \leq \delta$ .

We can define a path $\gamma \colon [|t|^{{{1}/{3}}} {K}/{\rho}, 1] \rightarrow V_0$ by

(42) \begin{equation} \gamma(s) = s^{{3}/{2}} \cdot p.\end{equation}

This path is chosen such that it begins in $D_0(|t|^{{{1}/{3}}}K)$ and moves outward along a ray emanating from 0 to reach $\gamma(1) = p$ . It can be checked that

\begin{equation*} \dot{\gamma}(s) = \rho \cdot \frac{\partial }{\partial r} \text{ and } r(\gamma(s)) = \rho \cdot s.\end{equation*}

The corresponding path in $V_t$ is $\tilde{\gamma} = \Phi_t \circ \gamma$ , and our goal is to estimate its length. We start with

\begin{equation*}\begin{aligned} |L_{\mathrm{co},t}(\widetilde{\gamma}) - L_{\mathrm{co},0}(\gamma)| &\leq \int_{|t|^{1/3} {K}/{\rho}}^1 \Bigg| |\dot{\gamma}|_{(\Phi_t)^* (g_{\mathrm{co},t})} - |\dot{\gamma}|_{g_{\mathrm{co},0}} \Bigg| \, ds \\ &\leq \int_{|t|^{1/3} {K}/{\rho}}^1 |(\Phi_t)^* g_{\mathrm{co},t} - g_{\mathrm{co},0}|_{g_{\mathrm{co},0}} |\dot{\gamma}|_{g_{\mathrm{co},0}} \, ds.\end{aligned}\end{equation*}

Using $|\dot{\gamma}|_{g_{\mathrm{co},0}} = \rho$ , $r(s) = s \cdot \rho$ and (CO SR II), we get

\begin{equation*} |L_{\mathrm{co},t}(\widetilde{\gamma}) - L_{\mathrm{co},0}(\gamma)| \leq C \int_{|t|^{1/3} {K}/{\rho}}^1 |t| r^{-3}(s) \rho \, ds \leq C \bigg( \frac{|t|^{1/3}}{K^2} - \frac{|t|}{\rho^2} \bigg).\end{equation*}

We can also check that

\begin{equation*} L_{\mathrm{co},0} (\gamma) = \int_{|t|^{1/3} {K}/{\rho}}^1 \rho \, ds = \rho - |t|^{1/3} K.\end{equation*}

By the triangle inequality and the above estimates, we conclude

\begin{equation*} L_{\mathrm{co},t}(\widetilde{\gamma}) \leq C( \rho + |t|^{1/3} K^{-2}).\end{equation*}

Since $|t| \leq \delta^3 K^{-3}$ and $\rho \leq \delta$ , we conclude that

(43) \begin{equation} d_{\mathrm{co},t}(q, D_t(\beta_{t,|t|^{{1}/{3}}K})) \leq C \delta.\end{equation}

Combining this with our diameter bound (41) for $D_t(\beta_{t,|t|^{{{1}/{3}}}K})$ , we get

(44) \begin{equation} \text{diam}_{\mathrm{co},t} ( D_t(\beta_{t,\delta})) \leq C \delta, \quad 0 \lt |t| \leq \frac{\delta^3}{K^3},\end{equation}

which is the desired diameter bound for the Calabi–Yau metrics on the local model $(V_t,g_{\mathrm{co},t})$ .

4.4.3 Bounds for the Fu–Li–Yau metrics.

On the smoothings, the Fu–Li–Yau metrics are only close to scaled Candelas–de la Ossa metrics instead of being exactly equal to them. Owing to this, we require a version of the diameter bound (44) for the Fu–Li–Yau metric. This will follow by virtue of the estimate (13).Consider a curve $\gamma$ on the disc $D_t(\beta_{t,\delta})$ . We compare the length of this path $\gamma$ with respect to the Fu–Li–Yau metric and to a scaled Candelas–de la Ossa metric:

\begin{equation*} |L_{\mathrm{FLY},t}(\gamma) - \sqrt{c} \cdot L_{\mathrm{co},t}(\gamma)| \leq \frac{1}{\sqrt{c}} \int_{0}^1 |g_{\mathrm{FLY},t} - c \cdot g_{\mathrm{co},t}|_{g_{\mathrm{co},t}} \cdot |\dot{\gamma}|_{g_{\mathrm{co},t}} \, ds.\end{equation*}

Using (13), and recognizing that the $0<\delta<1$ and $K \gg 1$ appearing here can be chosen such that $\beta_{t,\delta}$ is smaller than the $\delta$ appearing in (13) for all $0 \lt |t| \leq ({\delta}/{K})^3$ , we have

\begin{equation*} |L_{\mathrm{FLY},t}(\gamma) - \sqrt{c} \cdot L_{\mathrm{co},t}(\gamma)| \leq C |t|^{{2}/{3}} \, L_{\mathrm{co},t}(\gamma).\end{equation*}

Thus,

\begin{equation*} L_{\mathrm{FLY},t}(\gamma) \leq C(|t|^{{2}/{3}}+1) \, L_{\mathrm{co},t}(\gamma).\end{equation*}

It then follows that

\begin{equation*} \text{diam}_{\mathrm{FLY},t} (D_t(\beta_{t,\delta})) \leq C (|t|^{{2}/{3}} +1) \text{diam}_{\mathrm{co},t} ( D_t(\beta_{t,\delta})).\end{equation*}

Combining this with (44), we have the following.

Lemma 9. For $\delta \gt 0$ ,

(45) \begin{equation} \text{diam}_{\mathrm{FLY},t} (D_t(\beta_{t,\delta})) \leq C \delta (|t|^{{2}/{3}}+1), \end{equation}

for all $0 \lt |t| \leq ({\delta}/{K})^3$ .

4.4.4 Applying the main lemma.

Using the diameter estimates, we prove an analog of Lemma 8 in the case of the smoothings for the Candelas–de la Ossa and Fu–Li–Yau metrics.

Proof. As was the case in Lemma 8, the first condition holds as long as $\delta \lt {\epsilon}/{2}$ . Using (44) or (45), we see that for all $0 \lt |t| \leq ({\delta}/{K})^3$ , we have the estimate

\begin{equation} \text{diam}_{\mathrm{co},t} ( D_t(\beta_{t,\delta})) \leq C \delta \end{equation}

for some uniform constant $C \gt 0$ . The result follows.

Applying our lemma then gives convergence of our metrics on the smoothings.

• – Convergence of the local models $(D_t(\beta_{t,1}), d_{\mathrm{co},t}) \rightarrow (D_0(1), d_{\mathrm{co},0})$ .

  We have only one ODP singularity s for this case. Using our diameter estimate (44), we have that for $\epsilon \gt 0$ , we can pick $\overline{G} = D_0(\delta)$ , $K_t = D_0(({|t|}/{2})^{{{1}/{3}}})$ and $C_t = D_t(|t|^{{{1}/{3}}})$ for sufficiently small $\delta \gt 0$ and t such that Lemma 5 applies with the maps $\Phi_t$ .

• – Convergence of the global balanced metrics $(X_t,d_{\mathrm{FLY},t}) \rightarrow (X_0, d_{\mathrm{FLY},0})$ .

  Here, we use the diameter estimate (45) instead. For $\epsilon \gt 0$ , we can again pick $\overline{G_i} = D_0(\delta_i)$ , $K_{i,t} = D_0(({|t|}/{2})^{{{1}/{3}}})$ and $C_{i,t} = D_t(|t|^{{{1}/{3}}})$ for sufficiently small $\delta_i \gt 0$ and t around each singularity $s_i$ . As such, we can apply the lemma with the maps $\Phi_t$ .

• – Convergence of the global Hermitian Yang–Mills metrics $(X_t,d_{H_t}) \rightarrow (X_0,d_{H_0})$ .

  Similar to the case of the small resolutions, we use estimate (14), which is

  \begin{equation} C^{-1} \cdot g_{\mathrm{FLY},t} \leq H_t \leq C \cdot g_{\textrm{FLY,t}} \end{equation}
  on the local sets $D_0(\delta_i)$ around the singularities $s_i$ . Set $K_{i,t} = D_0(({|t|}/{2})^{{{1}/{3}}})$ and $C_{i,t} = D_t(|t|^{{{1}/{3}}})$ . Lemma 10 then gives that, for $\epsilon \gt 0$ , there exists $\delta_i \gt 0$ and $t_0 \gt 0$ such that for all $0 \lt |t| \lt t_0$ ,
  \begin{equation} \text{diam}_{H_0} (D_0(\delta_i)) \lt \epsilon, \quad \text{diam}_{H_t} (D_t(\beta_{t,\delta_i})) \lt \epsilon. \end{equation}
  Applying Lemma 5 using the maps $\Phi_t$ proves the result.

Combining these results with the analogous results for the small resolution found at the end of §4.3.2, we obtain Theorem 4.1.