Projective duality and a Chern-Mather involution

TL;DR

This paper reveals that projective duality preserves linear relations among Chern-Mather classes, leading to an explicit involution on Chow groups with applications to classical formulas, singular varieties, and Euclidean distance degree computations.

Contribution

It introduces an explicit involution on Chow groups that encodes the effect of projective duality on Chern-Mather classes, extending classical formulas to singular varieties.

Findings

Linear relations among Chern-Mather classes are preserved by duality.

An explicit involution on Chow groups encodes duality effects.

Applications include generalized Plücker formulas and Euclidean distance degree calculations.

Abstract

We observe that linear relations among Chern-Mather classes of projective varieties are preserved by projective duality. We deduce the existence of an explicit involution on a part of the Chow group of projective space, encoding the effect of duality on Chern-Mather classes. Applications include Pl\"ucker formulae, constraints on self-dual varieties, generalizations to singular varieties of classical formulas for the degree of the dual and the dual defect, formulas for the Euclidean distance degree, and computations of Chern-Mather classes and local Euler obstructions for cones.