One can prove using an argument due to Donaldson that items (a) and (b) are equivalent. The point is that $\Omega_{X}$ uniquely determines $J_{X}$ because $J_{X}$ is determined by knowing the subspace $\Lambda^{1,0}_{J_{X}}\subset\Lambda^{1}_{\mathbb{C}}X$, and we have $\Lambda^{1,0}_{J_{X}}X={\rm ker}\,T$, where $T:\Lambda^{1}_{\mathbb{C}}X\to\Lambda^{n+1}_{\mathbb{C}}X$ is the $\mathbb{C}$-linear map defined by $T\alpha=\Omega_{X}\wedge\alpha$. See Lemma 2.14 in [CH13] for details.

Item (b) can be proved by following the steps of a similar estimate in the asymptotically conical case in Section 2.2 of [CH15]. Fix any background hermitian metric $g$ on $M$. Via $g$-orthogonal projection, the holomorphic normal bundle $L=N_{D}=T^{1,0}M|_{D}/T^{1,0}D$ is naturally isomorphic to the $g$-orthogonal complement $(T^{1,0}D)^{\perp}\subset T^{1,0}M$ as a $C^{\infty}$ complex line bundle, and the $g$-normal exponential map defines a diffeomorphism from a neighborhood of the zero section in $(T^{1,0}D)^{\perp}$ to a neighborhood of $D$ in $M$. Let $\Phi$ be the composition of these two maps. Then $\Phi$ is a diffeomorphism from a neighborhood of the zero section in $L$ to a neighborhood of $D$ in $M$, and the restriction of $\Phi$ to the zero section is ${\rm Id}_{D}$. Note that $\Phi$ is almost never holomorphic, but in generic situations $\Phi$ will be one of the “most holomorphic” diffeomorphisms between tubular neighborhoods of $D$ in $L$ and in $M$. In any case, $\Phi$ turns out to be good enough to obtain the asymptotics (3.13).

Fix a point on $D$. Let $(z_{1},\ldots,z_{n-1},w)$ be local holomorphic coordinates on $M$ centered at this point such that $D$ is locally cut out by $w=0$. Then $(z_{1},\ldots,z_{n-1},w)$ may also be viewed as local holomorphic coordinates on $L$ corresponding to the normal vector $w(\frac{\partial}{\partial w}+T^{1,0}D)\in L$ based at the point $(z_{1},\ldots,z_{n-1},0)\in D$. In these coordinates we may write

(3.16)		$\displaystyle\Omega_{X}$	$\displaystyle=\biggl(\frac{f(z)}{w}+g(z,w)\biggr)dz_{1}\wedge\ldots\wedge dz_{n-1}\wedge dw,$
(3.17)		$\displaystyle\Omega_{\mathcal{C}}$	$\displaystyle=\frac{f(z)}{w}dz_{1}\wedge\ldots\wedge dz_{n-1}\wedge dw,$

where $f,g$ are local holomorphic functions with $f(z)\neq 0$ for all $z$. In order to compare $\Phi^{*}\Omega_{X}$ to $\Omega_{\mathcal{C}}$, we define new $C^{\infty}$ complex coordinates $(z_{1}^{\prime},\ldots,z_{n-1}^{\prime},w)$ on $M$ by

(3.18)

$\displaystyle z_{i}^{\prime}(z,w)=z_{i}-a_{i}(z)w,\ \text{where}\ a_{i}(z)=\frac{\partial(z_{i}\circ\Phi)}{\partial w}\biggr|_{(z,0)}.$

Using the fact that $d\Phi$ is complex linear at $w=0$ and that $\Phi_{*}(\frac{\partial}{\partial w})=\frac{\partial}{\partial w}+\sum a_{i}(z)\frac{\partial}{\partial z^{i}}$ at $w=0$, it is easy to check that these new coordinates satisfy

(3.19)

$\displaystyle\Phi^{*}dz_{i}^{\prime}=dz_{i}\ \text{and}\ \Phi^{*}dw=dw\ \text{at}\ w=0.$

By Taylor expansion, it follows directly from this that

(3.20)

$\displaystyle\Phi^{*}z_{i}^{\prime}=z_{i}+A_{i}w^{2}+B_{i}w\overline{w}+C_{i}\overline{w}^{2}\ \text{and}\ \Phi^{*}w=w+Aw^{2}+Bw\overline{w}+C\overline{w}^{2}$

with smooth functions $A_{i},B_{i},C_{i}$ and $A,B,C$. We now express the coordinates $(z,w)$ in (3.16) in terms of $(z^{\prime},w)$ using (3.18), and then use (3.20) to compare $\Phi^{*}\Omega_{X}$ to $\Omega_{\mathcal{C}}$. The first step yields

(3.21)

$\displaystyle\Omega_{X}=\frac{f(z^{\prime})}{w}dz_{1}^{\prime}\wedge\ldots\wedge dz_{n-1}^{\prime}\wedge dw+\Upsilon\wedge dw,$

where $\Upsilon$ extends to a smooth complex $(n-1)$-form on a neighborhood of $D$ in $M$. Then

(3.22)

$\displaystyle\begin{split}&\Phi^{*}\Omega_{X}-\Omega_{\mathcal{C}}\\ =&\frac{(A^{\prime}+B^{\prime}\frac{\overline{w}}{w}+C^{\prime}\frac{\overline{w}^{2}}{w^{2}})(\Upsilon^{\prime}\wedge dw)+(A^{\prime\prime}+B^{\prime\prime}\frac{\overline{w}}{w}+C^{\prime\prime}\frac{\overline{w}^{2}}{w^{2}})(\Upsilon^{\prime\prime}\wedge d\overline{w})+(w\Theta^{\prime}+\overline{w}\Theta^{\prime\prime}+\frac{\overline{w}^{2}}{w}\Theta^{\prime\prime\prime})}{1+Aw+B\overline{w}+C\frac{\overline{w}^{2}}{w}}\\ &+\Phi^{*}\Upsilon\wedge(dw+w\phi^{\prime}+\overline{w}\phi^{\prime\prime}),\end{split}$

where $A^{\prime},B^{\prime},C^{\prime},A^{\prime\prime},B^{\prime\prime},C^{\prime\prime},\Upsilon^{\prime},\Upsilon^{\prime\prime},\Theta^{\prime},\Theta^{\prime\prime},\Theta^{\prime\prime\prime},\phi^{\prime},\phi^{\prime\prime}$ extend to smooth complex functions, $(n-1)$-forms, $n$-forms, and $1$-forms on a neighborhood of $D$ in $L$, respectively. The reason for writing the right-hand side of (3.22) in this way is that a smooth complex $n$-form is small with respect to $g_{\mathcal{C}}$ if it either contains an explicit factor of $w$ or $\overline{w}$ in front, or if it splits off a wedge factor of $dw$ or $d\overline{w}$. Unfortunately the right-hand side of (3.22) is not smooth at the divisor but all non-smooth terms are due to factors of $\overline{w}/w$, which satisfy the same estimates as smooth functions.

It remains to prove appropriate estimates on $|\nabla^{k}_{g_{\mathcal{C}}}F|_{g_{\mathcal{C}}}$ for all $k\geq 0$, where $F$ is either a smooth function on a neighborhood of $D$ in $L$, or $F=\overline{w}/w$. To begin, note that

(3.23)		$\displaystyle|\nabla_{g_{\mathcal{C}}}^{k}z_{i}|_{g_{\mathcal{C}}}$	$\displaystyle=O(1)\ \text{for all}\ k\geq 0,$
(3.24)		$\displaystyle|w|$	$\displaystyle=O(e^{-\frac{1}{2}z^{n}}),\ |\nabla_{g_{\mathcal{C}}}^{k}w|_{g_{\mathcal{C}}}=O(e^{-(\frac{1}{2}-\epsilon)z^{n}})\ \text{for all}\ k\geq 1,\epsilon>0,$
(3.25)		$\displaystyle|w^{-1}|$	$\displaystyle=O(e^{\frac{1}{2}z^{n}}),\ |\nabla_{g_{\mathcal{C}}}^{k}w^{-1}|_{g_{\mathcal{C}}}=O(e^{(\frac{1}{2}+\epsilon)z^{n}})\ \text{for all}\ k\geq 1,\epsilon>0.$

Here the bound $|z_{i}|=O(1)$ is clear, and the bounds $|w^{\pm 1}|=O(e^{\mp\frac{1}{2}z^{n}})$ follow from the definition of the moment map $z=(-{\log|\xi|_{h}^{2}})^{1/n}$ together with the fact that $|\xi|_{h}^{2}=|w|^{2}e^{-\phi}$ with $\phi$ independent of $w,\overline{w}$. The higher derivative bounds in (3.23)–(3.25) follow from these pointwise bounds by using elliptic estimates for holomorphic functions on a Kähler manifold of $C^{\infty}$ bounded geometry (these estimates apply here because $g_{\mathcal{C}}$ is Ricci-flat Kähler of bounded curvature). Note that the $\epsilon$-terms in (3.24)–(3.25) are necessary because the sup of $|w|$ over a $g_{\mathcal{C}}$-ball of radius 1 is $O(|w|^{1-\epsilon})$ for every $\epsilon>0$ but is not $O(|w|)$, unlike on a cylinder $\mathbb{C}^{*}_{w}\times D$ with model metric $|d\log w|^{2}+g_{D}$.

We now prove by induction that for all smooth functions $F$ on a neighborhood of $D$ in $L$,

(3.26)

$$|F|=O(1),\ |\nabla_{g_{\mathcal{C}}}^{k}F|_{g_{\mathcal{C}}}=O(e^{\epsilon z^{n}})\ \text{for all}\ k\geq 1,\epsilon>0.$$

Indeed, the pointwise bound is clear, and for $k\geq 1$ we apply $\nabla_{g_{\mathcal{C}}}^{k-1}$ to the expansion

(3.27)

$$dF=\frac{\partial F}{\partial w}dw+\frac{\partial F}{\partial\overline{w}}d\overline{w}+\sum_{i=1}^{n-1}\frac{\partial F}{\partial z_{i}}dz_{i}+\sum_{i=1}^{n-1}\frac{\partial F}{\partial\overline{z}_{i}}d\overline{z}_{i},$$

using the inductive hypothesis to control $\nabla_{g_{\mathcal{C}}}^{k-1}$ of the partials of $F$ on the right-hand side and using (3.23)–(3.24) to control $\nabla_{g_{\mathcal{C}}}^{k-1}$ of $dz_{i},d\overline{z}_{i},dw,d\overline{w}$. This proves (3.26). By using (3.24)–(3.25) we can then prove in a similar manner that

(3.28)

$\displaystyle\left|\frac{\overline{w}}{w}\right|=O(1),\ \left|\nabla^{k}_{g_{\mathcal{C}}}\left(\frac{\overline{w}}{w}\right)\right|_{g_{\mathcal{C}}}=O(e^{\epsilon z^{n}})\ \text{for all}\ k\geq 1,\epsilon>0.$

Taken together, (3.26) and (3.28) allow us to estimate all contributions to (3.22) in all $C^{k}$ norms with respect to $g_{\mathcal{C}}$, proving item (b).

To prove item (c), notice that in local coordinates $\xi=(z_{1},\ldots,z_{n-1},w)\in L$ as above,

(3.29)

$$\Phi^{*}|S|^{2}_{h_{M}}=(1+G)|\xi|^{2}_{h}=(1+G)|w|^{2}e^{-\phi}$$

with smooth real-valued locally defined functions $G$ and $\phi$, where $G$ vanishes at $w=0$ and $\phi$ does not depend on $w,\overline{w}$. Notice that $G=Fw+\overline{Fw}$ for some smooth complex-valued locally defined function $F$. This structure of the $\Phi^{*}J_{X}$-Kähler potential of $\Phi^{*}\omega_{X}$, together with (3.24), (3.26), and item (a), makes it possible to prove that for all $k\geq 0,\epsilon>0$,

(3.30)

$$|\nabla^{k}_{g_{\mathcal{C}}}(\Phi^{*}\omega_{X}-\omega_{\mathcal{C}})|_{g_{\mathcal{C}}}=O(e^{-(\frac{1}{2}-\epsilon)z^{n}}).$$

Similarly by Theorem 3.3 we get for some $\underline{\delta}>0$ depending on $\delta_{0}$ that for all $k\geq 0$,

(3.31)

$$|\nabla^{k}_{g_{\mathcal{C}}}(\Phi^{*}\omega_{TY}-\Phi^{*}\omega_{X})|_{g_{\mathcal{C}}}=O(e^{-\underline{\delta}z^{n/2}}).$$

This completes the proof of item (c). ∎