Some examples of non-massive Frobenius manifolds in singularity theory

Abstract

Let f , g : C 2 → C be two quasi-homogeneous polynomials. We compute the V -filtration of the restriction of f to any plane curve C t = g − 1 ( t ) and show that the Gorenstein generator d x ∧ d y / d g is a primitive form. Using results of A. Douai and C. Sabbah, we conclude that the base space of the miniversal unfolding of f t ≔ f | C t is a Frobenius manifold. At the singular fibre C 0 we obtain a non-massive Frobenius manifold.