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

TL;DR

This paper explores the Hodge theory of degenerations, focusing on vanishing cohomology and its role in the limiting mixed Hodge structure, with applications to singularities in algebraic geometry and mirror symmetry.

Contribution

It advances understanding of the weighted spectrum and vanishing cohomology for various hypersurface singularities and their impact on geometric compactifications and mirror symmetry.

Findings

Analysis of vanishing cohomology for isolated hypersurface singularities

Applications to singularities in KSBA and GIT compactifications

Insights into the structure of limiting mixed Hodge structures

Abstract

We study the weighted spectrum and vanishing cohomology for several classes of 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 on odd-dimensional hypersurfaces, $k$-log-canonical and $k$-rational singularities, and singularities with Calabi-Yau tail.