Introduction The early history of mirror symmetry has been told many times we will only summarize it briefly here. The story begins with the introduction of Calabi-Yau compactifications in string theory in 1985 [11]. The idea is that, since superstring theory requires a ten-dimensional space-time, one reconciles this with the observed universe by requiring (at least locally) that space-time take the form R1,3 × X, where R1,3 is usual Minkowski space-time and X is a very small six-dimensional Riemannian manifold. The desire for the theory to preserve the supersymmetry of superstring theory then leads to the requirement that X have SU(3) holonomy, i.e., be a Calabi-Yau manifold. Thus string theory entered the realm of algebraic geometry, as any non-singular projective threefold with trivial canonical bundle carries a metric with SU(3) holonomy, thanks to Yau’s proof of the Calabi conjecture [113]. This generated an industry in the string theory community devoted to produc- ing large lists of examples of Calabi-Yau threefolds and computing their invariants, the most basic of which are the Hodge numbers h1,1 and h1,2. In 1989, a rather surprising observation came out of this work. Candelas, Lynker and Schimmrigk [12] provided a list of Calabi-Yau hypersurfaces in weighted projective space which exhibited an obvious symmetry: if there was a Calabi-Yau threefold with Hodge numbers given by a pair (h1,1,h1,2), then there was often also one with Hodge numbers given by the pair (h1,2,h1,1). Independently, guided by certain observations in conformal field theory, Greene and Plesser [36] studied the quintic threefold and its mirror partner. If we let Xψ be the solution set in P4 of the equation x0 5 + · · · + x4 5 − ψx0x1x2x3x4 = 0 for ψ ∈ C, then for most ψ, Xψ is a non-singular quintic threefold, and as such, has Hodge numbers h1,1(Xψ) = 1, h1,2(Xψ) = 101. On the other hand, the group G = {(a0,...,a4)|ai ∈ μ5, 4 i=0 ai = 1} {(a, a, a, a, a)|a ∈ μ5} acts diagonally on P4, via (x0,...,x4) → (a0x0,...,a4x4). Here μ5 is the group of fifth roots of unity. This action restricts to an action on Xψ, and the quotient Xψ/G is highly singular. However, these singularities can be xi

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2011 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.