Some examples of non-massive Frobenius manifolds in Singularity Theory

Abstract

Let $f,g:\cc^2\ra\cc$ 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 $dx\wedge dy/dg$ is a primitive form. Using results of A. Douai and C. Sabbah, we conclude that 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.