Floer cohomology, multiplicity and the log canonical threshold

Abstract

Let $f$ be a polynomial over the complex numbers with an isolated singularity at $0$. We show that the multiplicity and the log canonical threshold of $f$ at $0$ are invariants of the link of $f$ viewed as a contact submanifold of the sphere.

This is done by first constructing a spectral sequence converging to the fixed-point Floer cohomology of any iterate of the Milnor monodromy map whose $E^1$ page is explicitly described in terms of a log resolution of $f$. This spectral sequence is a generalization of a formula by A’Campo. By looking at this spectral sequence, we get a purely Floer-theoretic description of the multiplicity and log canonical threshold of $f$.