Let $\mathbb{T}=\mathbb{R}^{3}/\Lambda$ be a $3$–torus for some lattice $\Lambda\simeq\mathbb{Z}^{3}$. Endow $\mathbb{T}$ with a flat metric $g_{\mathbb{T}}$.

Let $\tau\colon\thinspace\mathbb{T}\rightarrow\mathbb{T}$ be the standard involution $x\mapsto-x$ on $\mathbb{T}$ and denote by $q_{1},\dots,q_{8}$ its fixed points. For each $j=1,\dots,8$ choose a non-negative integer $m_{j}$.

Fix a $\tau$–symmetric configuration of further $2n$ distinct points $p_{1},\tau(p_{1}),\dots,p_{n},\tau(p_{n})$. Sometimes we will use the notation $-p_{i}$ for $\tau(p_{i})$. Denote by $\mathbb{T}^{\ast}$ the punctured torus

Finally choose integer weights $k_{1},\dots,k_{n}>0$ and assume the following balancing condition holds:

In particular, $n\leq\sum_{i=1}^{n}{k_{i}}\leq 16$.

For each $j=1,\dots,8$ let $\rho_{j}$ denote the distance function from the point $q_{j}$ with respect to $g_{\mathbb{T}}$. Similarly, by abuse of notation we let $\rho_{i}$ denote the distance function from $\pm p_{i}$ in $\mathbb{T}/\tau$. By restricting the branched double cover $\mathbb{T}\rightarrow\mathbb{T}/\tau$ to a sufficiently small ball centred at $\pm p_{i}$ in $\mathbb{T}/\tau$ we will also regard $\rho_{i}$ as the distance function on $\mathbb{T}$ from the point $p_{i}$ or $\tau(p_{i})$.

We look for a Dirac monopole $(h,\theta)$ on $\mathbb{T}^{\ast}$ with the following singular behaviour: $h$ is a harmonic function on $\mathbb{T}^{\ast}$ with prescribed singularities at the punctures

Fix once and for all a Dirac monopole $(h,\theta)$ amongst the ones produced by Proposition 4.3. Via the Gibbons–Hawking ansatz (3.3) we now use $(h,\theta)$ to construct (incomplete) hyperkähler metrics with a triholomorphic circle action with orbits of small length.

Fix a (small) positive number $\epsilon>0$ and define

The Gibbons–Hawking ansatz (3.3) yields a hyperkähler structure on $P|_{\mathcal{U}_{\epsilon}}$, where $\mathcal{U}_{\epsilon}$ is the open set of $\mathbb{T}^{\ast}$ where $h_{\epsilon}>0$. The hyperkähler triple $\bm{\underline{\omega}}^{\textup{gh}}_{\epsilon}$ and the induced metric $g^{\textup{gh}}_{\epsilon}$ are

Here $(\theta_{1},\theta_{2},\theta_{3})$ is a triple of closed $1$–forms on $\mathbb{T}$ such that $g_{\mathbb{T}}=\theta_{1}^{2}+\theta_{2}^{2}+\theta_{3}^{2}$. Note that the hyperkähler structure $\bm{\underline{\omega}}^{\textup{gh}}_{\epsilon}$ is $\tilde{\tau}$–invariant and therefore defines an induced hyperkähler structure on the quotient $M^{\textup{gh}}_{\epsilon}=\left(P|_{\mathcal{U}_{\epsilon}}\right)/\tilde{\tau}$. For ease of notation, we will denote the induced hyperkähler triple and metric on $M^{\textup{gh}}_{\epsilon}$ with the same symbols.

In the rest of the section we study the properties of the hyperkähler manifold $M^{\textup{gh}}_{\epsilon}$. We aim to (i) study the local structure of the metric $g^{\textup{gh}}_{\epsilon}$ close to the punctures, (ii) determine the set where $h_{\epsilon}>0$, and (iii) understand the limit of $g^{gh}_{\epsilon}$ as $\epsilon\rightarrow 0$.

The following asymptotic expansions for the harmonic function $h_{\epsilon}$ close to the punctures are standard. In the case of a fixed point of the involution $\tau$ the improved decay follows from the fact that linear harmonic functions on $\mathbb{R}^{3}$ are not $\mathbb{Z}_{2}$–invariant.

Since we chose $k_{i}>0$ for $i=1,\dots,n$ certainly a punctured neighbourhood of $\pm p_{i}$ is contained in the set $\mathcal{U}_{\epsilon}$ where $h_{\epsilon}>0$. As already mentioned, the Gibbons–Hawking metric can be extended by adding a single point to a smooth orbifold metric modelled on $\mathbb{C}^{2}/\mathbb{Z}_{k_{i}}$. The obvious way to smooth out such an orbifold singularity is to replace the “multiplicity” $k_{i}$ point $p_{i}$ with $k_{i}$ points each with weight $1$. Then the Gibbons–Hawking ansatz yields a smooth metric that is modelled on a rescaled Taub–NUT space in a neighbourhood of $p_{i}$. However we prefer to leave the freedom to choose $k_{i}>1$ so that we can consider configurations of punctures that “degenerate” as $\epsilon\rightarrow 0$ and see an $A_{k_{i}-1}$ ALF space appearing as a rescaled limit.

Next, we consider the structure of $g^{\textup{gh}}_{\epsilon}$ in a neighbourhood of $q_{j}$. By Lemma 4.7.(i) certainly $h_{\epsilon}$ is positive in a punctured neighbourhood of $q_{j}$ whenever $m_{j}>2$. In this case the Gibbons–Hawking metric on $M^{\textup{gh}}_{\epsilon}$ can be extended to a smooth orbifold metric with a singularity of the form $\mathbb{C}^{2}/\mathcal{D}_{m_{j}}$, where $\mathcal{D}_{m_{j}}$ is the binary dihedral group of order $4(m_{j}-2)$. In contrast with the previous case, there is no explicit way to remove this singularity, but for fixed $\epsilon>0$ one can imagine using the methods of [8, §2.4] to resolve this singularity by gluing in a rescaled ALE dihedral space. However, we are interested in the limit $\epsilon\rightarrow 0$ and in the next section we will directly glue in a $D_{m_{j}}$ ALF space to resolve this singularity.

Similarly, when $m_{j}=2$ one can choose $\epsilon$ sufficiently small so that $1+\epsilon\lambda_{j}>0$. Note that in this case $h_{\epsilon}$ and the $U(1)$–bundle $P$ are well defined at $q_{j}$. After quotienting by $\tilde{\tau}$, the Gibbons–Hawking metric $g^{\textup{gh}}_{\epsilon}$ becomes an orbifold metric modelled on $(\mathbb{R}^{3}\times S^{1})/\mathbb{Z}_{2}$. As before, the two orbifold singularities could be resolved by (i) fixing $\epsilon>0$ and gluing in two copies of the Eguchi–Hanson metric, or (ii) letting $\epsilon\rightarrow 0$ and gluing in a single copy of a $D_{2}$ ALF metric. We will follow the second approach.

It remains to study the case when $m_{j}=0,1$ for some $j$. Assume this is the case for $j=1,\dots,k$ for some $1\leq k\leq 8$. Since $\sum_{j=1}^{8}{m_{j}}=16-\sum_{i=1}^{n}{k_{i}}\leq 16-n$, note that $k\geq 1$ as soon as $n\geq 1$ or $n=0$ and $m_{j}\neq 2$ for some $j$, i.e. in every case except for the usual Kummer construction. The case $m_{j}=0,1$ is “bad” in the sense that $h_{\epsilon}\rightarrow-\infty$ as $\rho_{j}\rightarrow 0$.

In other words, by choosing $\epsilon$ small enough we can assume that $h_{\epsilon}>0$ outside an arbitrarily small neighbourhood of the points $q_{1},\dots,q_{k}$ where $m_{j}=0,1$.

Finally, we consider the limit $\epsilon\rightarrow 0$.