Isolated hypersurface singularities, spectral invariants, and quantum cohomology

Abstract We study the relation between isolated hypersurface singularities (e. g. , ADE) and the quantum cohomology ring by using spectral invariants, which are symplectic measurements coming from Floer theory. We prove, under the assumption that the quantum cohomology ring is semi-simple, that (1) if the smooth Fano variety degenerates to a Fano variety with an isolated hypersurface singularity, then the singularity has to be an A m A₌ -singularity, (2) if the symplectic manifold contains an A m A₌ -configuration of Lagrangian spheres, then there are consequences for the Hofer geometry, and that (3) the Dehn twist reduces spectral invariants.