Linear free divisors and Frobenius manifolds

TL;DR

This paper investigates linear functions on fibrations with linear free divisors, analyzing their Gauss-Manin systems and establishing Frobenius manifold structures on their base spaces, revealing new geometric insights.

Contribution

It demonstrates the existence of a primitive, homogeneous form and constructs Frobenius manifolds from semi-universal unfoldings of these functions.

Findings

Existence of a primitive, homogeneous form for the Gauss-Manin system

Construction of Frobenius manifold structures on base spaces

Link between linear free divisors and Frobenius manifolds

Abstract

We study linear functions on fibrations whose central fibre is a linear free divisor. We analyse the Gauss-Manin system associated to these functions, and prove the existence of a primitive and homogenous form. As a consequence, we show that the base space of the semi-universal unfolding of such a function carries a Frobenius manifold structure.