Variational Bihamiltonian Cohomologies and Integrable Hierarchies I: Foundations

This series of papers is devoted to the study of Virasoro symmetries of deformations of the Principal Hierarchy associated with a semisimple Frobenius manifold. The main tool we use is a generalization of the bihamiltonian cohomology called the variational bihamiltonian cohomology. In the present paper, we give its definition and compute the associated cohomology groups that will be used in our study of Virasoro symmetries. As an application of the variational bihamiltonian cohomology theory, we classify conformal bihamiltonian structures with semisimple hydrodynamic limits.