Collapsing of $AL{H}^{\ast }$-Gravitational Instantons

Abstract

We showed that a sequence of ALH*-gravitational instantons from pairs consisting of a weak del Pezzo surface and a smooth anti-canonical divisor towards a large complex structure limit introduced by Collins, Jacobs and the first author collapsing to a punctured plane with a special Kahler metric, which can be viewed as a non-compact version of the collapsing result of Gross-Wilson. We provide a partial compactification of the moduli space of pointed ALH*-gravitational instantons with respect to the pointed Gromov-Hausdorff topology and locally is a polyhedron complex.

arXiv comment23 Pages, Comments are welcome. Section 3.3 is added to clarify the cases when singular fibres converge at the large complex structure limit. Some corrections in Section 5 are made