Special Kähler Metrics on Complex Line Bundles and the Geometry of K3-Surfaces

In this article we continue studying the Ricci-flat Riemannian metrics that were constructed in [1]. On closer examination it turned out that they possess a number of remarkable properties; in particular, they have the holonomy group SU(2), so presenting special Kähler metrics. The metrics of holonomy SU(2) are interesting because of their applications in mathematical physics. In superstring theory and M -theory there appear compact manifolds with special holonomy groups. Moreover, if we admit the presence of physically isolated singularities then it suffices to study asymptotically flat metrics on the normal bundles of these singularities. Thus, we arrive at the problem of studying asymptotically locally Euclidean metrics with special holonomies on bundles over orbifolds. One of the most topical examples of special Kähler metrics is the Eguchi– Hanson metric [2] on the cotangent bundle T S of the standard two-dimensional sphere (without singularities). The Eguchi–Hanson metric has played an important role in studying special holonomy groups. Namely, Page [3] proposed a description of the space of special Kähler metrics on a K3-surface in which the Eguchi–Hanson metric plays the role of an “elementary brick.” More exactly, represent a K3-surface using Kummer’s construction; i.e., consider the involution of the flat torus T 4 which arises from the central symmetry of the Euclidean space R. Factorizing, we obtain an orbifold with 16 singular points whose neighborhoods look like C/Z2. Blowing up the resulting orbifold in a neighborhood of each singular point, we obtain a K3-surface. Topologically, the construction of blowing up a singular point of the form C/Z2 is carried out as follows: We have to delete the singularity and identify its neighborhood with the space of the spherical bundle in T S without the zero fiber S. Page proposed to consider a metric on T S which is homothetic to the Eguchi–Hanson metric with a sufficiently small homothety coefficient so that the metric on the boundary of the glued spherical bundle becomes arbitrarily close to a flat metric. After that we need to deform slightly the metric on the torus so as to obtain a smooth metric on a K3-surface with holonomy SU(2). A simple evaluation of the degrees of freedom in the process of this operation demonstrates that we obtain a 58-dimensional family of metrics which agrees with the well-known