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 be the solution set in P4 of the equation x0 5 + · · · + x4 5 ψx0x1x2x3x4 = 0 for ψ C, then for most ψ, 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
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