The natural $S^{1}$-action on $\mathcal{C}$ obviously preserves ${\bm{\omega}}_{\mathcal{C}}$, so ${\bm{\omega}}_{\mathcal{C}}$ must be given by the Gibbons-Hawking construction. It suffices to determine the hyperkähler moment map $\pi=(\pi_{1},\pi_{2},\pi_{3})$ and the potential $V$. We already know that the $\omega_{\mathcal{C}}$-moment map is given by $\pi_{1}=(-{\log|\xi|_{h}^{2}})^{1/2}=z$, and it is easy to check that the $\Omega_{\mathcal{C}}$-moment map $\pi_{2}+i\pi_{3}$ equals the bundle projection $p:\mathcal{C}\to E$ followed by the Abel-Jacobi isomorphism $E\to\mathbb{C}/\text{\emph{periods}}$, $x\mapsto\int_{x_{0}}^{x}i\Omega_{E}$, for an arbitrary but fixed basepoint $x_{0}\in E$. Also, $V^{-1}$ is the norm-squared of the Killing field, so that

The Calabi construction provides us with an explicit realization $\mathfrak{M}=\mathcal{C}$ of the total space of the $S^{1}$-bundle and with a specific choice of connection form $\theta$ given by the Chern connection of $L$ with respect to $h$. The diffeomorphism equivalence to the model ${\rm Nil}_{b}^{3}(\epsilon,\tau)\times(0,\infty)$ with connection form $\theta_{b}$ (after rotating $\Omega_{\mathcal{C}}$ to $e^{i\alpha}\Omega_{\mathcal{C}}$ if necessary) follows from the discussion before Remark 2.4. ∎

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

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

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

By Taylor expansion, it follows directly from this that

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

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

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

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 $L$,

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

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

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,

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 $k\geq 0,\epsilon>0$,

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

This completes the proof of item (c). ∎

By Proposition 3.1, the Calabi space $\mathcal{C}$ is diffeomorphic to $\Nil^{3}_{b}\times(0,\infty)$ in such a way that the Calabi metric $g_{\mathcal{C}}$ becomes the Gibbons-Hawking metric (2.19). Ignoring this diffeomorphism, we have a $2$-form triple ${\bm{\phi}}$ and a $1$-form triple ${\bm{\psi}}$ such that $\partial_{z}\,\lrcorner\,{\bm{\phi}}=0$, $\partial_{z}\,\lrcorner\,{\bm{\psi}}=0$, and

Since $\bm{\omega}_{TY},\bm{\omega}_{\mathcal{C}}$ are closed, it follows that

Now we define the $1$-form triple

Thanks to (3.32), this integral exists and satisfies (3.34). Note that this is not completely obvious because the $1$-form basis $dx,dy,dt-xdy$ on ${\rm Nil}^{3}_{b}$ is not parallel with respect to $g_{\mathcal{C}}$. However, this effect is absorbed by the $\epsilon$ in (3.34) because all error terms are at worst polynomial in $z$. Property (3.33) now follows in a standard manner by using (3.36). ∎