Hodge symmetry and Lefschetz theorems for singular varieties

Abstract

We prove new results concerning the topology and Hodge theory of singular varieties. A common theme is that concrete conditions on the complexity of the singularities, from a number of different perspectives, are closely related to the symmetries of the Hodge-Du Bois diamond. We relate this to the theory of rational homology manifolds, and characterize these among low-dimensional varieties with rational singularities.

38 pages. First part of a two-paper replacement of the original submission. The content about Q-factoriality has been expanded and moved to arXiv:2508.17748. This part contains new results on rational homology manifolds and bounds for the local cohomological defect