Adjacent Singularities, TQFTs, and Zariski's Multiplicity Conjecture

TL;DR

This paper presents a new proof of Zariski's multiplicity conjecture for isolated hypersurface singularities using TQFT and symplectic cobordisms, offering a novel approach that also recovers Varchenko's theorem.

Contribution

The authors introduce a TQFT-based method utilizing Floer cohomology and symplectic cobordisms to prove Zariski's conjecture, providing an alternative to previous proofs.

Findings

New proof of Zariski's multiplicity conjecture for isolated hypersurface singularities.

Construction of a chain map on Floer cochains using symplectic cobordisms.

Recovery of Varchenko's theorem through spectral sequence analysis.

Abstract

We give a new proof of Zariski's multiplicity conjecture in the case of isolated hypersurface singularities; this was first proved by de Bobadilla-Pe\l ka \cite{BobadillaPelka}. Our proof uses the TQFT structure of fixed-point Floer cohomology and the fact that adjacent singularities produce symplectic cobordisms between the Milnor fibrations of the singularities. The key technical result is to construct a chain map on Floer cochains using the cobordism and as a last step, apply a spectral sequence of McLean \cite{McLeanLog}. This last step allows us to also recover a theorem of Varchenko \cite{varchenko}.