In this subsection, we will define $3$-dimensional Heisenberg nilmanifolds. Recall the $3$-dimensional Heisenberg group is

To define a Heisenberg nilmanifold, let us define a co-compact group action on $H(1,\mathbb{R})$. First, we define a lattice $\Lambda\equiv\epsilon\mathbb{Z}\langle 1,\tau\rangle\subset\mathbb{R}^{2}_{x,y}=\mathbb{C}$ by choosing

which is generated by

Then immediately $A=\Area(\mathbb{R}_{x,y}^{2}/\Lambda)=\epsilon^{2}\tau_{2}$. For $b\in\mathbb{Z}_{+}$, the Heisenberg nilmanifold $\Nil^{3}_{b}(\epsilon,\tau)$ of degree $b$ is the quotient of $H(1,\mathbb{R})$ by the left action generated by

Note that these transformations are

The forms

are a basis of left-invariant $1$-forms.

It is clear that $\Nil^{3}_{b}$ is the total space of a degree $b$ circle fibration

The following result will be needed later in Proposition 6.6 to determine the Betti numbers of $\mathcal{M}$.

For $b\in\mathbb{Z}_{+}$, we define the Heisenberg nilmanifold $\Nil^{3}_{-b}$ to be the quotient of $H(1,\mathbb{R})$ by the action generated by

Note that the generated action is conjugate to the previous action by the mapping $(x,y,t)\mapsto(-x,-y,-t)$. The forms

are a basis of left-invariant $1$-forms.

Consider a $2$-torus $\mathbb{T}^{2}$ with a flat metric of area $A$ and let $U=\mathbb{T}^{2}_{x,y}\times\mathbb{R}_{z>0}$, where we have fixed a choice of an orthogonal frame $\{e_{1},e_{2}\}$ on $\mathbb{T}^{2}$ such that $g_{\mathbb{T}^{2}}=A(e_{1}^{2}+e_{2}^{2})$. Let $V(z)=\frac{2\pi bz}{A}$ for a positive integer $b>0$. Fixing a connection form $\theta$ such that $d\theta=\frac{2\pi b}{A}{\rm dvol}_{\mathbb{T}^{2}}$, the corresponding hyperkähler Gibbons-Hawking metric $g$ is given by

The Gibbons-Hawking space $(\mathfrak{M},g)$ has one complete end as $z\rightarrow\infty$ and one incomplete end as $z\rightarrow 0$. Moreover, for each $z_{0}>0$, the level set $\{z=z_{0}\}$ is a Heisenberg nilmanifold $\Nil^{3}_{b}(\epsilon,\tau)$ with a $z_{0}$-dependent left-invariant metric, where $\tau$ denotes the modulus of our flat $2$-torus in the upper half-plane and $A=\epsilon^{2}\tau_{2}$ as in Section 2.1. Making the substitution $z=(3/2)s^{2/3}$, and then scaling appropriately, the Gibbons-Hawking metric $g$ takes the form

In this form, it is easy to see that the volume growth is $\sim s^{4/3}$ and that $|{\Rm}|\sim s^{-2}$ as $s\rightarrow\infty$. One can also show using the Chern-Gauss-Bonnet formula that the $L^{2}$ norm of $\Rm$ is finite.

Note that if we had instead taken $b=0$, the Gibbons-Hawking metric would be the product of $\mathbb{R}$ with a flat $3$-torus, i.e., the type of geometry known as ALH geometry in the literature.

View the flat torus $\mathbb{T}^{2}$ as an elliptic curve $E$ of modulus $\tau$ with respect to the complex structure $J$ defined by $Je_{1}=e_{2}$. Then there is exactly one $g$-parallel complex structure $J_{0}$ on the total space $\mathfrak{M}$ that makes the projection map to $E$ holomorphic. This choice of complex structure realizes $\mathfrak{M}$ as an open subset of the total space of a degree $b$ holomorphic line bundle $L$ over $E$ (more precisely, as a tubular neighborhood of the zero section of $L$ with the zero section removed). The Ricci-flat Kähler form with respect to $J_{0}$ is then given by (see Proposition 3.1)

where $h$ is a hermitian metric on $L$ whose curvature form is a multiple of the flat Kähler form on $E$, and where the tautological section $\xi$ cuts out the zero section of $L$. This is an example of the Calabi construction, and we call the corresponding metric a Calabi model metric of degree $b$. We will discuss this construction (in all dimensions) in more detail in Section 3. In complex dimension $2$, the Gibbons-Hawking ansatz actually recovers the Calabi ansatz for all degree $b$ holomorphic line bundles over $E$. Indeed, any two such line bundles differ by a degree 0 line bundle and $\Pic^{0}(E)=H^{1}(E,\mathcal{O}_{E})/H^{1}(E,\mathbb{Z})$; on the other hand, the gauge equivalence classes of flat connections are parametrized by $H^{1}(E,\mathbb{R})/H^{1}(E,\mathbb{Z})$.

A very convenient property for our purposes is that if we do change the connection $1$-form $\theta$ by a flat connection, then the associated Gibbons-Hawking metric (2.19) actually only changes by a diffeomorphism (although this diffeomorphism necessarily breaks the $S^{1}$-bundle structure on $\mathfrak{M}$). To see this, fix any choice of $\theta$ such that $d\theta=\frac{2\pi b}{A}\dvol_{E}$. Note that we can always arrange by parallel transport in the $z$-direction that the $dz$-component of $\theta$ vanishes. Then we only need to consider the case that $\theta$ gets changed by the pullback (under the projection $\mathfrak{M}\stackrel{{\scriptstyle\pi}}{{\to}}U\to E$) of a parallel $1$-form $\eta$ on $E$. Write $\eta=v\,\lrcorner\,\dvol_{E}$, where $v$ is a parallel vector field on $E$. Let $\hat{v}$ be the $\theta$-horizontal lift of $v$ to $\mathfrak{M}$ and let $\hat{f}_{s}$ be the $1$-parameter group of diffeomorphisms generated by $\hat{v}$, which covers a $1$-parameter group of translations on $E$. Then

using the fact that $d\hat{f}_{s}|_{x}(\hat{v}|_{x})=\hat{v}|_{\hat{f}_{s}(x)}$ for all $x\in\mathfrak{M}$. Thus,

This shows that any two possible choices of $\theta$ with vanishing $dz$-component differ by the translation action of $E$ on itself, lifted to $\mathfrak{M}$, and then the corresponding hyperkähler metrics (2.19) are clearly isometric as well. We note that with the particular choice $\theta=\theta_{b}$ from (2.12), these diffeomorphisms can be written down explicitly as follows: for all $p,q\in\mathbb{R}$, the mapping

descends to a diffeomorphism of ${\rm Nil}^{3}_{b}(\epsilon,\tau)_{x,y,t}\times\mathbb{R}_{z}$ which satisfies

The Calabi construction provides the model at infinity of the Tian-Yau metrics in our context. These metrics will also be studied in more detail in Section 3. Note that every Tian-Yau metric comes with its own preferred choice of a connection form $\theta$ determined by the Chern connection of the normal bundle of the compactifying divisor. The above gauge fixing construction then allows us to choose a new coordinate system at infinity that identifies this $\theta$ with the standard connection form $\theta_{b}$ (see the proof of Proposition 3.1 and also equation (6.6)).

The neck region in our gluing construction is given by a doubly-periodic analog of the Ooguri-Vafa metric. To construct this metric using the Gibbons-Hawking ansatz, we first need to construct a Green’s function $V_{\infty}$ on $(\mathbb{T}^{2}\times\mathbb{R},g_{0})$, where $\mathbb{T}^{2}$ is any flat $2$-torus. We also need to determine the asymptotics of $V_{\infty}$ as $|z|\to\infty$ in order to ensure that the neck matches the Calabi model space from Section 2.2 on both ends.

It is known that the flat cylinder $\mathbb{T}^{2}\times\mathbb{R}$ is parabolic, namely, it does not admit a positive Green’s function. In fact, Cheng and Yau proved that any complete non-compact Riemannian manifold with $\Vol(B_{R}(p))\leq CR^{2}$ must be parabolic. See theorem 1 and corollary 1 in [CY75].

We now construct a particular sign-changing Green’s function $V_{\infty}$. Fix a point $p$ on $\mathbb{T}^{2}\times\mathbb{R}$ with $z(p)=0$. For $R>0$ let $V_{R}$ denote the unique function on $\mathbb{T}^{2}\times[-R,R]$ satisfying

The normalization of the right-hand side is precisely chosen in such a way that $V_{R}=\frac{1}{2r}+O(1)$ near $p$, where $r$ denotes $g_{0}$-distance to $p$. Thus, the $4$-dimensional Gibbons-Hawking metric associated with $V_{R}$ (or $V_{R}+C$ for any constant $C$) extends smoothly across $p$; cf. Example 2.1.

By the maximum principle, one can see that $V_{R}$ is an increasing family as $R\to\infty$. Define

Then it follows from the parabolicity of $\mathbb{T}^{2}\times\mathbb{R}$ that $C_{R}\to\infty$ as $R\to\infty$. By a result of Li and Tam (see theorem 1 in [LT87]), $V_{R}(x,y,z)-C_{R}$ converges to a function $V_{\infty}(x,y,z)$ uniformly on compact subsets of $(\mathbb{T}^{2}\times\mathbb{R})\setminus\{p\}$. Moreover, $V_{\infty}\leq 0$ on the complement of $B_{1}(p)$ and

Notice also that $V_{\infty}$ is symmetric in $z$ by construction.

It is straightforward to use superposition to extend the above construction to the case of multiple poles. Precisely, we have the following corollary.

In Section 6, we will use $V_{\infty}$ to construct a potential $V_{\beta}$ which is positive in a large region, and such that $[\frac{1}{2\pi}*dV_{\beta}]\in H^{2}(U,\mathbb{Z})$ is a integral class, which will then be used to define our neck metric.