In mathematics, mirror symmetry is a conjectural relationship between certain Calabi–Yau manifolds and a constructed "mirror manifold". The conjecture allows one to relate the number of rational curves on a Calabi-Yau manifold (encoded as Gromov–Witten invariants) to integrals from a family of varieties (encoded as period integrals on a variation of Hodge structures). In short, this means there is a relation between the number of genus algebraic curves of degree on a Calabi-Yau variety and integrals on a dual variety . These relations were original discovered by Candelas, De la Ossa, Green, and Parkes in a paper studying a generic quintic threefold in as the variety and a construction from the quintic Dwork family giving . Shortly after, Sheldon Katz wrote a summary paper outlining part of their construction and conjectures what the rigorous mathematical interpretation could be. Constructing the mirror of a quintic threefold Originally, the construction of mirror manifolds was discovered through an ad-hoc procedure. Essentially, to a generic quintic threefold there should be associated a one-parameter family of Calabi-Yau manifolds which has multiple singularities. After blowing up these singularities, they are resolved and a new Calabi-Yau manifold was constructed. which had a flipped Hodge diamond. In particular, there are isomorphisms but most importantly, there is an isomorphism where the string theory (the A-model of ) for states in is interchanged with the string theory (the B-model of ) having states in . The string theory in the A-model only depended upon the Kahler or symplectic structure on while the B-model only depends upon the complex structure on . Here we outline the original construction of mirror manifolds, and consider the string-theoretic background and conjecture with the mirror manifolds in a later section of this article. Complex moduli Recall that a generic quintic threefold in is defined by a homogeneous polynomial of degree . This polynomial is equivalently described as a global section of the line bundle . Notice the vector space of global sections has dimension but there are two equivalences of these polynomials. First, polynomials under scaling by the algebraic torus (non-zero scalers of the base field) given equivalent spaces. Second, projective equivalence is given by the automorphism group of , which is dimensional. This gives a dimensional parameter space since , which can be constructed using Geometric invariant theory. The set corresponds to the equivalence classes of polynomials which define smooth Calabi-Yau quintic threefolds in