Moduli Space of Calabi–Yau Manifolds

We present an accessible account of the local geometry of the parameter space of Calabi–Yau manifolds. It is shown that the parameter space decomposes, at least locally, into a product with one factor the space of parameters of the complex structure and the other a complex extension of the parameter space of the Kähler class. It is also shown that each of these spaces is itself a Kähler manifold and is moreover a Kähler manifold of restricted type. There is a remarkable symmetry in the intrinsic structures of the two parameter spaces and the relevance of this to the conjectured existence of mirror manifolds is discussed. The two parameter spaces behave differently with respect to modular transformations and it is argued that the role of quantum corrections is to restore the symmetry between the two types of parameters so as to enforce modular invariance.