arXiv subjects
Publications and source records attributed to Claus Hertling.
We give an overview on the tt*-geometry defined for isolated hypersurface singularities and tame functions via Brieskorn lattices. We discuss nilpotent orbits in this context, as well as classifying spaces of Brieskorn lattices and (limits of) period maps.
The distribution of the spectral numbers of an isolated hypersurface singularity is studied in terms of the Bernoulli moments. These are certain rational linear combinations of the higher moments of the spectral numbers. They are related to the generalized Bernoulli polynomials. We conjecture that their signs are alternating and prove this in many cases. One motivation for the Bernoulli moments comes from the comparison with compact complex manifolds.
The existence of universal unfoldings of certain germs of meromorphic connections is established. This is used to prove a general construction theorem for Frobenius manifolds. A particular case is Dubrovin's theorem on semisimple Frobenius manifolds. Another special case starts with variations of Hodge structures. This case is used to compare two constructions of Frobenius manifolds, the one in singularity theory and the Barannikov-Kontsevich construction. For homogeneous polynomials which give Calabi-Yau hypersurfaces certain Frobenius submanifolds in both constructions are isomorphic.
The base space of a semiuniversal unfolding of a hypersurface singularity carries a rich geometry. By work of K. Saito and M. Saito is 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. In the first part (chapters 2-5) 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.
Bolibruch's examples of representations of pi_1(P^1-finitely many points) which are not realizable by Fuchsian differential systems are adapted to curves of higher genus.
A formula for the variance of the spectrum of a quasihomogeneous singularity is proved, using the G-function of a semisimple Frobenius manifold.
Manifolds with a commutative and associative multiplication on the tangent bundle are called F-manifolds if a unit field exists and the multiplication satisfies a natural integrability condition. They are studied here. They are closely related to discriminants and Lagrange maps. Frobenius manifolds are F-manifolds. As an application a conjecture of Dubrovin on Frobenius manifolds and Coxeter groups is proved.
We establish a new universal relation between the Lie bracket and $\circ$-multiplication of tangent fields on any Frobenius (super)manifold. We use this identity in order to introduce the notion of ``weak Frobenius manifold'' which does not involve metric as part of structure. As another application, we show that the powers of an Euler field generate (a half of) the Virasoro algebra on an arbitrary, not necessarily semisimple, Frobenius supermanifold.