Reeb orbits and the minimal discrepancy of an isolated singularity

Abstract

Let A be an affine variety inside a complex N dimensional vector space which has an isolated singularity at the origin. The intersection of A with a very small sphere turns out to be a contact manifold called the link of A. Any contact manifold contactomorphic to the link of A is said to be Milnor fillable by A. If the first Chern class of our link is torsion then we can assign an invariant of our singularity called the minimal discrepancy, which is an important invariant in birational geometry. We define an invariant of the link up to contactomorphism using Conley-Zehnder indices of Reeb orbits and then we relate this invariant with the minimal discrepancy. As a result we show that the standard contact 5 dimensional sphere has a unique Milnor filling up to normalization proving a conjecture by Seidel.