
## 3. (Q)ALX in higher dimensions

Motivated by the modular interpretation of ALG-gravitational instantons as Hitchin’s $L^{2}$-metrics on moduli spaces of parabolic Higgs bundles, it is desirable to find a characterization of the geometry of a class of metrics that would include all Hitchin’s $L^{2}$-metrics in higher dimensions. This leads to the important yet vague question: What is (Q)ALX in higher dimensions?

Many known examples should be included in the resulting characterization, for instance, Hilbert schemes of points in $\mathbb{C}^{2}$, as well as more general quiver varieties for comet or squid-shaped quivers that are expected to be QALE.

There are two important aspects to be considered and contrasted when pursuing this problem, namely, the resulting volume growth at infinity versus the characterization of precise asymptotic models.
1. ### Existence of growth rate

(Gromov; quotation?)

#### Problem 3.1.

[Song Sun] The volume rate of a (Q)ALX metric in dimensions greater than 4 is not well-defined in general. What is its suitable generalization to higher dimensions?
• ### Gap theorems

The a priori expected volume growths for a non-compact complete $4k$-hyperkähler manifold are $r^{4k}, r^{3k}, r^{2k}$, and $r^{k}$. Bando–Kasue–Nakajima [MR1001844] have proved that an $n$-dimensional Ricci-flat manifold with curvature in $L^{n/2}$ has maximal (i.e., Euclidean) volume growth if it is asymptotic to $\mathbb{R}^{n}/G$ for a finite subgroup $G\in \mathrm{SO}(n)$. For $n=4$ this is $L^2$ as required for a gravitational instanton. Taking this work as an important motivational example leads to the following question.

#### Problem 3.2.

[Hartmut Weiss] Assume the volume growth of an $n$-dimensional Ricci-flat manifold (with suitable constraints) is strictly smaller than $O\left(r^{n}\right)$. Are there fractional rates of the form $n-\epsilon$, or is it necessarily $O\left(r^{n-1}\right)$? In the case of a $4k$-dimensional hyperkähler manifold: is the volume growth necessarily $O\left(r^{3k}\right)$?
Notice that the hyperkähler case $k=1$ is a theorem of Minerbe [MR2778451].
• ### Asymptotic models

One possible definition of a QALG $4k$-dimensional hyperkähler manifold suggested by Sergey A. Cherkis is simply to require its volume growth to be equal to $O\left(r^{2k}\right)$. It is then important to understand the type of potential asymptotic model constraints resulting from this definition.

#### Problem 3.3.

[Hartmut Weiss] (i) It is possible to derive an asymptotic model from the previous growth requirement? If so, it should be some (singular) fibration with half-dimensional flat fiber, like a semi-flat metric.

(i) What are the possible models at infinity for monopole moduli paces, and $\star$-shaped quiver varieties with loops?

Cite this as: AimPL: Geometry and physics of ALX metrics in gauge theory, available at http://aimpl.org/geomphysalx.