Floer Cohomology, Multiplicity and the Log Canonical Threshold

TL;DR

This paper links algebraic invariants of a polynomial with isolated singularities to Floer cohomology, providing a new Floer-theoretic perspective on multiplicity and log canonical thresholds.

Contribution

It constructs a spectral sequence connecting Floer cohomology to algebraic invariants, generalizing A'Campo's formula for isolated singularities.

Findings

Spectral sequence converges to fixed point Floer cohomology of Milnor monodromy

Explicit E^1 page description in terms of log resolution

Floer theoretic description of multiplicity and 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.