Hodge theory of degenerations, (III): a vanishing-cycle calculus for non-isolated singularities

TL;DR

This paper develops a calculus for vanishing cycles in the Hodge theory of degenerations, focusing on non-isolated singularities with one-dimensional singular loci, including complex examples like log-canonical and surface singularities.

Contribution

It introduces a concrete computational framework for non-isolated singularities in Hodge theory, extending previous isolated singularity results.

Findings

Computed vanishing cycles for non-isolated singularities with one-dimensional loci

Analyzed specific surface singularities and their Hodge structures

Applied methods to singular 5-folds in Feynman integral contexts

Abstract

We continue our study of the Hodge theory of degenerations, Part I of which covered consequences of the Decomposition Theorem and Part II of which concerned geometric applications in the isolated singularity case. The focus here in Part III is on concrete computations in the case of non-isolated singularities, particularly those for which the singular locus has dimension one. These examples are significantly more involved than in the previous parts, and include $k$-log-canonical singularities, several specific surface singularities (both slc and non-slc), and certain singular 5-folds arising in the study of Feynman integrals.