Deformation of Singularities and the Homology of Intersection Spaces

TL;DR

This paper investigates how the cohomology of intersection spaces behaves under smooth deformations of complex hypersurface singularities, revealing stability properties and connections to mixed Hodge structures.

Contribution

It demonstrates the stability of intersection space cohomology under deformations except possibly in the middle degree, linking trivial monodromy to this stability and establishing a mixed Hodge structure.

Findings

Cohomology of intersection spaces is stable under smooth deformations in all degrees except possibly the middle.

Trivial monodromy action on the Milnor fiber's cohomology corresponds to stability in the middle degree.

Rational cohomology of intersection spaces admits a mixed Hodge structure compatible with that of the hypersurface.

Abstract

While intersection cohomology is stable under small resolutions, both ordinary and intersection cohomology are unstable under smooth deformation of singularities. For complex projective algebraic hypersurfaces with an isolated singularity, we show that the first author's cohomology of intersection spaces is stable under smooth deformations in all degrees except possibly the middle, and in the middle degree precisely when the monodromy action on the cohomology of the Milnor fiber is trivial. In many situations, the isomorphism is shown to be a ring homomorphism induced by a continuous map. This is used to show that the rational cohomology of intersection spaces can be endowed with a mixed Hodge structure compatible with Deligne's mixed Hodge structure on the ordinary cohomology of the singular hypersurface.