Periods and Picard-Fuchs Equations

One way to describe a complex projective variety X is to give equations cutting out X from an ambient projective space. At times however, there is a more intrinsic way of describing the complex structure of X if we know the underlying topological manifold. The additional data are called the periods of X and may be viewed as integrals of holomorphic forms on X over cycles in X.

In principle, the equations of X determine the periods of X. In practice, it is hard to compute the periods given an equation and vice versa. We will talk about how one can determine the periods of X by first computing the periods of a more favorable X' and deforming X' to X, keeping track of the change in periods via the so called Picard-Fuchs equations.