Deformations of complex structures on Riemann surfaces and integrable structures of Whitham type hierarchies

Abstract

We obtain variational formulas for holomorphic objects on Riemann surfaces with respect to arbitrary local coordinates on the moduli space of complex structures. These formulas are written in terms of a canonical object on the moduli space which corresponds to the pairing between the space of quadratic differentials and the tangent space to the moduli space. This canonical object satisfies certain commutation relations which appear to be the same as the ones that emerged in the integrability theory of Whitham type hierarchies. Driven by this observation, we develop the theory of Whitham type hierarchies integrable by hydrodynamic reductions as a theory of certain differential-geometric objects. As an application we prove that the universal Whitham hierarchy is integrable by hydrodynamic reductions.