Isolated hypersurface singularities, spectral invariants, and quantum cohomology (Q6547189)

Degeneration is a theme that originates in classical algebraic geometry and is still actively studied in the context of various modern topics such as the minimal model program, Kähler-Einstein metrics, mirror symmetry, and the SYZ conjecture. Its importance in symplectic topology was noticed by \textit{V. I. Arnold} [Prog. Math. 133, 99--103 (1995; Zbl 0970.53043)] and \textit{S. K. Donaldson} [in: Mathematics: frontiers and perspectives. Providence, RI: American Mathematical Society (AMS). 55--64 (2000; Zbl 0958.57027)], especially that Lagrangians can appear as vanishing cycles.\N\NThe author investigates the relation between isolated hypersurface singularities, e.g., those of ADE type, and the quantum cohomology ring, by using spectral invariants, which are symplectic measurements coming from Floer theory. He proves, 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 \(A_m\)-singularity, (2) if the symplectic manifold contains an \(A_m\)-configuration of Lagrangian spheres, then there are consequences for the Hofer geometry, and that (3) a Dehn twist reduces spectral invariants.