Hermitian metrics on F-manifolds

TL;DR

This paper reviews the construction of hermitian metrics on F-manifolds from tt* geometry, introduces a new canonical metric, and shows it induces near hyperbolicity near irreducible points, with potential applications to singularity theory.

Contribution

It introduces a new canonical hermitian metric on F-manifolds and clarifies the relationships between existing notions, enhancing understanding of their geometric properties.

Findings

The new hermitian metric makes the manifold almost hyperbolic near irreducible points.

The construction applies to base spaces of universal unfoldings of hypersurface singularities.

Clarifies the logical connections between different notions in tt* geometry.

Abstract

An $F$-manifold is complex manifold with a multiplication on the holomorphic tangent bundle with a certain integrability condition. Important examples are Frobenius manifolds and especially base spaces of universal unfoldings of isolated hypersurface singularities. This paper reviews the construction of hermitian metrics on $F$-manifolds from $tt^*$ geometry. It clarifies the logic between several notions. It also introduces a new {\it canonical} hermitian metric. Near irreducible points it makes the manifold almost hyperbolic. This holds for the singularity case and will hopefully lead to applications there.