The Euclidean distance degree of smooth complex projective varieties

TL;DR

This paper derives multiple formulas for calculating the Euclidean distance degree of smooth complex projective varieties, connecting it to various characteristic classes and topological invariants.

Contribution

It introduces new formulas relating the ED degree to Chern, Segre, Milnor, and Chern-Schwartz-MacPherson classes, including a simple Euler characteristic-based formula.

Findings

Formulas expressed via characteristic classes

Euler characteristic formula for ED degree

Applicability to nonsingular projective varieties

Abstract

We obtain several formulas for the Euclidean distance degree (ED degree) of an arbitrary nonsingular variety in projective space: in terms of Chern and Segre classes, Milnor classes, Chern-Schwartz-MacPherson classes, and an extremely simple formula equating the Euclidean distance degree of X with the Euler characteristic of an open subset of X.