, giving a moduli space of Calabi-Yau quintics. Now, using Serre duality and the fact each Calabi-Yau manifold has trivial canonical bundle , the space of deformations has an isomorphism with the part of the Hodge structure on . Using the Lefschetz hyperplane theorem the only non-trivial cohomology group is since the others are isomorphic to . Using the Euler characteristic and the Euler class, which is the top Chern class, the dimension of this group is . This is because Using the Hodge structure we can find the dimensions of each of the components. First, because is Calabi-Yau, so giving the Hodge numbers , hence giving the dimension of the moduli space of Calabi-Yau manifolds. Because of the Bogomolev-Tian-Todorov theorem, all such deformations are unobstructed, so the smooth space is in fact the moduli space of quintic threefolds. The whole point of this construction is to show how the complex parameters in this moduli space are converted into Kähler parameters of the mirror manifold. Mirror manifold There is a distinguished family of Calabi-Yau manifolds called the Dwork family. It is the projective family over the complex plane . Now, notice there is only a single dimension of complex deformations of this family, coming from having varying values. This is important because the Hodge diamond of the mirror manifold has Anyway, the family has symmetry group acting by Notice the projectivity of is the reason for the condition The associated quotient variety has a crepant resolution given by blowing up the singularities giving a new Calabi-Yau manifold with parameters in . This is the mirror manifold and has where each Hodge number is . Ideas from string theory In string theory there is a class of models called non-linear sigma models which study families of maps where is a genus algebraic curve and is Calabi-Yau. These curves are called world-sheets and represent the birth and death of a particle as a closed string. Since a string could split over time into two strings, or more, and eventually these strings will come together and collapse at the end of the lifetime of the particle, an algebraic curve mathematically represents this string lifetime. For simplicity, only genus 0 curves were considered originally, and many of the results popularized in mathematics focused only on this case. Also, in physics terminology, these theories are heterotic string theories because they have supersymmetry that comes in