Mirror Symmetry and Integral Variations of Hodge Structure Underlying One Parameter Families of Calabi-Yau Threefolds

This proceedings note introduces aspects of the authors’ work relating mirror symmetry and integral variations of Hodge structure. The emphasis is on their classification of the integral variations of Hodge structure which can underly families of Calabi-Yau threefolds over P \ {0, 1,∞} with b = 4, or equivalently h = 1, and the related issues of geometric realization of these variations. The presentation parallels that of the first author’s talk at the BIRS workshop. 1. Integral structures in mirror symmetry 1.1. The first examples. Since its introduction to the mathematical community through the seminal papers of Greene-Plesser [GP] and Candelas-de la Ossa-Green-Parkes [CdOGP], Mirror Symmetry has been the source of the most persistently rich and subtle novel mathematics yet to emerge from the study of string dualities. One of its key features is its resistance to rigorous mathematical definition and even more to being described within any single traditional mathematical setting. Research in mathematics related to mirror symmetry is thus driven by the goal of discerning deeper, more complete, and purely mathematical avatars of this very physical duality. For example, one of the earliest mirror symmetric mathematical predictions was that a “mirror pair” of Calabi-Yau threefolds X and X̃ should have the property that their Hodge numbers “mirror” one another, i.e., that h(X) = h(X̃) and h(X) = h(X̃). The first proposal for such a pair was introduced in [GP] where it was applied to Calabi-Yau threefolds presented as hypersurfaces in a weighted projective space. The construction involves the operation of quotienting by a finite group (or “orbifolding”). The simplest of Calabi-Yau threefold hypersurfaces, the generic quintic hypersurface in P, often denoted by P[5], has h = 1 (coming from the polarization class) and h = 101 (corresponding to 101 complex structure moduli). This last 1991 Mathematics Subject Classification. Primary 14D07, 14J32; Secondary 14M25, 19L64.