The main result of this paper gives a new gluing construction in which a family of hyperkähler metrics on a $\K 3$ surface collapses to a unit interval and generically the collapse happens along a $3$-dimensional Heisenberg nilmanifold (i.e., a nontrivial $S^{1}$-bundle over $\mathbb{T}^{2}$). Part of our motivation was an attempt to understand the hyperkähler metric degenerations corresponding to Type II complex structure degenerations of polarized $\K 3$ surfaces. A guiding example is when we have a family of quartic $\K 3$ surfaces $Z_{t}$ in $\mathbb{C}P^{3}$ defined by the equation $tq+f_{1}f_{2}=0$, where $q$ is a general quartic and $f_{1}$ and $f_{2}$ are general quadrics. So the general fiber is a smooth $\K 3$ surface while the central fiber is a union of two quadric surfaces $X_{1}$ and $X_{2}$, intersecting transversally along an elliptic curve defined by $f_{1}=f_{2}=0$. We would like to understand the behavior of the Ricci-flat metrics on $Z_{t}$ in the cohomology class of $2\pi c_{1}(\mathcal{O}(1)|_{Z_{t}})$ as $t$ tends to zero.

In general, given a del Pezzo surface $M$ and a smooth anti-canonical curve $D\subset M$, Tian-Yau proved in [TY90] the existence of a hyperkähler metric on $M\setminus D$, with interesting asymptotic geometry at infinity. Namely, outside a compact set the manifold is diffeomorphic to $N\times[0,\infty)$, where $N$ is an $S^{1}$-bundle over $D$ of degree $d=c_{1}(X)^{2}$, and the metric is modeled on a doubly-warped product so that as we move towards infinity the $S^{1}$-fibers shrink in size while the base torus $D$ expands. The volume growth rate of the hyperkähler metric is $4/3$ and the curvature decays quadratically. We call such a hyperkähler metric a Tian-Yau metric throughout this paper, and for more precise details we refer to Section 3. The proof in [TY90] uses the Calabi ansatz in a neighborhood of infinity and then solves a Monge-Ampère equation, so it is not a priori clear whether these metrics are unique or canonical in a suitable sense. Nevertheless they provide candidates for the bubble limits of the degeneration that we would like to understand.

Motivated by the above Type II degeneration picture, our basic idea was to glue together two Tian-Yau metrics to obtain hyperkähler $\K 3$ surfaces. However, an easy topological consideration shows that one cannot naively glue the ends of the two Tian-Yau metrics together to even match the topology of a $\K 3$ surface. Geometrically, even though the end of a Tian-Yau metric is toplogically cylindrical, the metric itself is not. So we need to construct a neck region that approximates the Tian-Yau ends on both sides.

A key novel ingredient of this paper is exactly to construct such a transition region. It is an incomplete hyperkähler $4$-manifold that can be viewed as a doubly-periodic cousin of the Ooguri-Vafa metric. Recall that the Ooguri-Vafa metric is the metric arising from the Gibbons-Hawking ansatz applied to a harmonic function on $S^{1}\times\mathbb{R}^{2}$ with a pole on $S^{1}\times\{0\}$; or equivalently, a harmonic function on $\mathbb{R}\times\mathbb{R}^{2}$, periodic in the first variable, and with poles on $\mathbb{Z}\times\{0\}\subset\mathbb{R}\times\{0\}$. Our neck metric is instead constructed by applying the Gibbons-Hawking ansatz to a harmonic function on the flat cylinder $\mathbb{T}^{2}\times\mathbb{R}$ with finitely many poles (which we call the monopole points) in $\mathbb{T}^{2}\times\mathbb{R}$. This is equivalent to a harmonic function on $\mathbb{R}^{2}\times\mathbb{R}$, doubly periodic in the first and second variables, and with poles on lattices. For more details of this construction we refer to Section 2. Here we point out that in analogy with the Ooguri-Vafa case, the resulting metric is incomplete because $\mathbb{T}^{2}\times\mathbb{R}$ is parabolic, hence admits no globally positive harmonic functions; moreover, the two ends of this neck metric indeed match up closely with the ends of Tian-Yau metrics.

Our main theorem says that it is in fact possible to “glue together” two Tian-Yau metrics with a suitable neck region as above to construct families of Ricci-flat metrics on $\K 3$ with non-trivial nilpotent collapsing structure.

For the precise definition of a $3$-dimensional Heisenberg nilmanifold, see Section 2.1. These are $S^{1}$-bundles over $\mathbb{T}^{2}$, and thus they have a degree which is only well-defined up to sign. However, if one specifies a projection to an oriented $\mathbb{T}^{2}$, then the degree is a well-defined integer. We denote by $\Nil_{b}^{3}$ a $3$-dimensional nilmanifold of degree $b$, where we will always have a certain projection to an oriented $\mathbb{T}^{2}$ in mind. Let $t_{0}=0$, $t_{m+1}=1$, and let $d_{j}$ be the degree of a nilpotent fiber $\Nil_{d_{j}}^{3}$ on the interval $(t_{j}+\epsilon,t_{j+1}-\epsilon)$, $j=0,\dots,m$, with $d_{0}=b_{-}$ and $d_{m}=-b_{+}$. The following Domain Wall Crossing Theorem describes the possible jumps of the degrees of the nilmanifolds upon crossing the singular regions.