Compactifications of Moduli Spaces Inspired by Mirror Symmetry

The study of moduli spaces by means of the period mapping has found its greatest success for moduli spaces of varieties with trivial canonical bundle, or more generally, varieties with Kodaira dimension zero. Now these moduli spaces play a pivotal rôle in the classification theory of algebraic varieties, since varieties with nonnegative Kodaira dimension which are not of general type admit birational fibrations by varieties of Kodaira dimension zero. Since such fibrations typically include singular fibers as well as smooth ones, it is important to understand how to compactify the corresponding moduli spaces (and if possible, to give geometric interpretations to the boundary of the compactification). Note that because of the possibility of blowing up along the boundary, abstract compactifications of moduli spaces are far from unique. The hope that the period mapping could be used to construct compactifications of moduli spaces was given concrete expression in some conjectures of Griffiths [22, §9] and others in the late 1960's. In particular, Griffiths conjectured that there would be an analogue of the Satake-Baily-Borel compactifications of arithmetic quotients of bounded symmetric domains, with some kind of " minimality " property among compactifications. Although there has been much progress since [22] in understanding the behavior of period mappings near the boundary of moduli, compactifications of this type have not been constructed, other than in special cases. In the case of algebraic K3 surfaces, the moduli spaces themselves are arithmetic quotients of bounded symmetric domains, so each has a minimal (Satake-Baily-Borel) compactification. In studying the moduli spaces for K3 surfaces of low degree in the early 1980's, Looijenga [31] found that the Satake-Baily-Borel compactification needed to be blown up slightly in order to give a good geometric interpretation to the boundary. He introduced a class of compactifications, the semi-toric compactifications, which includes the ones with a good geometric interpretation. In higher dimension, the moduli spaces are not expected to be arithmetic quotients of symmetric domains, so different techniques are needed. The study of these moduli spaces has received renewed attention recently, due to the discovery by theoretical physicists of a phenomenon called " mirror symmetry ". One of the predictions of mirror symmetry is that the moduli space for a variety with trivial canonical bundle, which parameterizes the possible complex structures on the underlying differentiable manifold, should also serve as