Hodge theory of degenerations, (II): vanishing cohomology and geometric applications.

We study the mixed spectrum and vanishing cohomology for several classes of (isolated and non-isolated) hypersurface singularities, and how they contribute to the limiting mixed Hodge structure of a smoothing. Applications are given to several types of singularities arising in KSBA and GIT compactifications and mirror symmetry, including nodes, $k$-log-canonical singularities, singularities with Calabi-Yau tail, normal-crossing degenerations, slc surface singularities, and the $J_{k,\infty}$ series.