Collapsing of $ALH^*$-gravitational instantons

By the uniformization theorem of [24], any $ALH^*$-gravitational instanton $(X,g)$ can be compactified to a weak del Pezzo surface $Y$ as a complex manifold by adding a smooth anti-canonical divisor $D$. A sequence of $ALH^*$-gravitational instantons $X_i$ converges to a large complex structure limit if the corresponding sequence of weak del Pezzo surfaces $Y_i$ converges to a weak del Pezzo surface $Y_0$ and the corresponding sequence of anti-canonical divisors $D_i$ converges to a nodal curve [8]. In this paper, we show that when $(X_i,g_i)$ converges to a large complex structure limit, then $X_i$ with a suitable scaling of the metric $g_i$ pointed Gromov-Hausdorff converges to a punctured plane with a special K\"ahler metric. The result can be viewed as a non-compact version of the collapsing result of Gross-Wilson [19]. Moreover, we provide a partial compactification of the moduli space of pointed $ALH^*$-gravitational instantons with respect to the pointed Gromov-Hausdorff topology.