Minimal Discrepancy of Isolated Singularities and Reeb Orbits

Let A be an affine variety inside a complex N dimensional vector space which either has an isolated singularity at the origin or is smooth 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 0 then we can assign an invariant of our singularity called the minimal discrepancy, which is an important invariant in birational geometry. We relate the minimal discrepancy with indices of certain Reeb orbits on our link. As a result we show that the standard contact 5 dimensional sphere has a unique Milnor filling up to normalization.