Fundamental Groups of Links of Isolated Singularities

Starting with Grothendieck’s proof of the local version of the Lefschetz hyperplane theorems [Gro68], it has been understood that there are strong parallels between the topology of smooth projective varieties and the topology of links of isolated singularities. This relationship was formulated as one of the guiding principles in the monograph [GM88, p. 26]: “Philosophically, any statement about the projective variety or its embedding really comes from a statement about the singularity at the point of the cone. Theorems about projective varieties should be consequences of more general theorems about singularities which are no longer required to be conical.” The aim of this note is to prove the following, which we consider to be a strong exception to this principle.