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