On the Hard Lefschetz property of stringy Hodge numbers

TL;DR

This paper proves the Hard Lefschetz property for stringy Hodge numbers in specific classes of singular projective varieties, aligning with known counterexamples in higher dimensions.

Contribution

It establishes the Hard Lefschetz property for stringy Hodge numbers in certain singular varieties, extending previous understanding and providing explicit examples.

Findings

Hard Lefschetz holds for varieties with mild isolated singularities

Validates the property for projective threefolds with Gorenstein canonical singularities

Provides a hypersurface singularity example where the property fails in higher dimensions

Abstract

For projective varieties with a certain class of 'mild' isolated singularities and for projective threefolds with arbitrary Gorenstein canonical singularities, we show that the stringy Hodge numbers satisfy the Hard Lefschetz property. This result fits nicely with a 6-dimensional counterexample of Mustata and Payne for the Hard Lefschetz property for stringy Hodge numbers in general. We also give such an example, ours is a hypersurface singularity.