Gauss–Manin systems

We discuss the theory of Gauss–Manin systems (or Picard–Fuchs differential systems) from the point of view of the theory of logarithmic differential forms. Consider a family of algebraic varieties depending on a parameter. An important example of such a family is an algebraic function depending on the coefficients of an algebraic equation. An integral of periods associated with an algebraic variety from this family, can be interpreted as a multivalued function depending on deformation parameters. In several important cases, such integrals satisfy a Pfaffian system (i. e., a Gauss–Manin system) whose coefficients are meromorphic differential forms having logarithmic poles along a discriminant set in the space of deformation parameters.