Topology of the Milnor fibrations of polar weighted homogeneous polynomials

Abstract

Let P be a 2-variable polar weighted homogeneous polynomial and let $$F_t$$ be a deformation of P which is also a polar weighted homogeneous polynomial. If $$|F_{t}|$$ is a Morse function on the orbit space of the $$S^1$$ -action, then the handle decomposition obtained by this Morse function induces a round handle decomposition of the Milnor fibration of $$F_t$$ . In the present paper, we describe a round handle decomposition of the Milnor fibration of $$F_t$$ concretely and give the number of round handles by the number of positive and negative components of the links of singularities appearing before and after the deformation. We also give a formula of characteristic polynomials of these singularities by using the decomposition of the monodromy of the Milnor fibration induced by a round handle decomposition.