The critical space for orthogonally invariant varieties

TL;DR

This paper investigates the geometry of invariant varieties under orthogonal group actions, identifying a critical subspace where closest points and critical points of distance functions lie, and applies this to compute Euclidean Distance degrees.

Contribution

It introduces the concept of the critical space for invariant varieties and generalizes previous results to broader classes including flag varieties.

Findings

Critical points of distance functions lie in the critical space.

Closest points to a given vector are contained in the critical space.

Euclidean Distance degree of complete flag varieties is computed.

Abstract

Let $q$ be a nondegenerate quadratic form on $V$. Let $X\subset V$ be invariant for the action of a Lie group $G$ contained in $SO(V,q)$. For any $f\in V$ consider the function $d_f$ from $X$ to $C$ defined by $d_f(x)=q(f-x)$. We show that the critical points of $d_f$ lie in the subspace orthogonal to ${\mathfrak g}\cdot f$, that we call critical space. In particular any closest point to $f$ in $X$ lie in the critical space. This construction applies to singular t-ples for tensors and to flag varieties and generalizes a previous result of Draisma, Tocino and the author. As an application, we compute the Euclidean Distance degree of a complete flag variety.