Symplectic Monodromy at radius 0 and equimultiplicity of families of hypersurfaces with constant Milnor number

Abstract: We prove the Zariski multiplicity conjecture for families of isolated hypersurface singularities. For it we show how to construct a symplectic monodromy ”at radius 0” with very special dynamical properties, which is symplectically isotopic by radius variation to the usual symplectic monodromy at positive small radius. For this we use a hybrid construction employing log-geometry (Kato-Nakayama spaces) and tropical geometry. In particular our construction provides a smooth atlas in the topological space used by A’Campo for his study of monodromy zeta function and Lefschetz numbers, and provides an alternate construction of special symplectic monodromy representatives due to McLean which is better suited for the study of families and degenerations. Then we use a slight generalization of a spectral sequence in Floer Homology (due to McLean) to recover multiplicity, and properties of invariance of Floer homology along symplectic isotopies to prove its constancy in Milnor number constant families.