Splayed divisors and their Chern classes

TL;DR

This paper introduces new characterizations of splayed divisors, explores their impact on Chern classes of associated sheaves, and proposes a conjectural identity relating these classes, verified in specific cases.

Contribution

It provides novel characterizations of splayedness and investigates their influence on Chern classes, including a conjecture connecting different Chern class notions.

Findings

New characterizations of splayedness via ideal properties and module isomorphisms.

Analysis of how splayedness affects Chern classes of logarithmic sheaves.

Verification of a conjectural identity for Chern-Schwartz-MacPherson classes in specific scenarios.

Abstract

We obtain several new characterizations of splayedness for divisors: a Leibniz property for ideals of singularity subschemes, the vanishing of a `splayedness' module, and the requirements that certain natural morphisms of modules and sheaves of logarithmic derivations and logarithmic differentials be isomorphisms. We also consider the effect of splayedness on the Chern classes of sheaves of differential forms with logarithmic poles along splayed divisors, as well as on the Chern-Schwartz-MacPherson classes of the complements of these divisors. A postulated relation between these different notions of Chern class leads to a conjectural identity for Chern-Schwartz-MacPherson classes of splayed divisors and subvarieties, which we are able to verify in several template situations.