Pull-back Morphisms for Reflexive Differential Forms

TL;DR

This paper develops a method to pull back reflexive differential forms through morphisms between complex varieties, even over singular loci, ensuring naturality and uniqueness, and extends to stratifications.

Contribution

It introduces a universal, natural construction for pull-back of reflexive differential forms on Kawamata log terminal varieties, including over singular loci.

Findings

Constructed a pull-back map for differential forms on singular varieties.

Proved the pull-back map is unique and natural.

Extended results to stratifications of singularities.

Abstract

Let f : X -> Y be a morphism between normal complex varieties, and assume that Y is Kawamata log terminal. Given any differential form, defined on the smooth locus of Y, we construct a "pull-back form" on X. The pull-back map obtained by this construction is O_Y-linear, uniquely determined by natural universal properties and exists even in cases where the image of f is entirely contained in the singular locus of the target variety Y. One relevant setting covered by the construction is that where f is the inclusion (or normalisation) of the singular locus of Y. As an immediate corollary, we show that differential forms defined on the smooth locus of Y induce forms on every stratum of the singularity stratification. The same result also holds for many Whitney stratifications.