Convolution of Picard-Fuchs equations (Q2102512)

Explicit expressions for Picard-Fuchs equations (or Gauss-Manin connections in a general context) defined after families of algebraic varieties are usually huge even if the corresponding family is simple. The authors propose a new method which uses the internal fibration structure of algebraic varieties. It involves only solving linear equations, and is a generalization of the classical convolution of solutions of Fuchsian differential equations and Deligne's work on cohomology with coefficients in a local system. The work is motivated by the increasing need for explicit expressions of Picard-Fuchs equations in topological string theory and in particular in the B-model of mirror symmetry. The authors determine explicit generators for a cohomology group constructed from a solution of a Fuchsian linear differential equation and describe its relation with cohomology groups with coefficients in a local system. In the parametrized case, this yields an algorithm which computes new Fuchsian differential equations from those depending on multi-parameters.