Geometry , Frobenius Manifolds , Their Connections , and the Construction for Singularities

The base space of a semiuniversal unfolding of a hypersurface singularity carries a rich geometry. By work of K. Saito and M. Saito it can be equipped with the structure of a Frobenius manifold. By work of Cecotti and Vafa it can be equipped with tt∗ geometry if the singularity is quasihomogeneous. tt∗ geometry generalizes the notion of variation of Hodge structures. In the second part of this paper (chapters 6–8) Frobenius manifolds and tt∗ geometry are constructed for any hypersurface singularity, using essentially oscillating integrals; and the intimate relationship between polarized mixed Hodge structures and this tt∗ geometry is worked out. It builds on the first part (chapters 2–5). There tt∗ geometry and Frobenius manifolds and their relations are studied in general. To both of them flat connections with poles are associated, with distinctive common and different properties. A frame for a simultaneous construction is given.