Lê's vanishing polyhedron for a family of mixed functions

Abstract

We study real analytic isolated singularities of type f:false(Cn+m+1,0false)→false(double-struckC,0false) with ffalse(z,wfalse)=gfalse(zfalse)+∑i=1m+1hifalse(wi,w¯ifalse), where g is holomorphic and each hi is a mixed polynomial, with z=(z1,⋯,zn) and w=(w1,⋯,wm+1). We construct a Lê's vanishing polyhedron for f, which describes the degeneration of its Milnor fiber Ff to the singular fiber. Then we prove that Ff is homotopy equivalent to the join Fg∗Fh1∗⋯∗Fhm+1, where Fg is the Milnor fiber of g and Fhi is the Milnor fiber of hi. This implies that Ff has the homotopy type of a bouquet of spheres Sn+m. So we can define the Milnor number μ(f) as the number of spheres in that bouquet, as in the complex setting.

Erratum

“…Between September 2019 and February 2020, twenty‐one research articles 1–21 were published in the Bulletin of the London Mathematical Society with incorrect copyright statements. The copyright in these articles is stated as belonging to the London Mathematical Society; however, that is incorrect and the copyright remains with the authors of those articles.…”