Abstract
Let q be a nondegenerate quadratic form on V. Let X ⊂ V be invariant for the action of a Lie group G contained in SO(V,q). For any f ∈ V consider the function df from X to mathbb C defined by df(x) = q(f − x). We show that the critical points of df 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.
Highlights
Let V be a complex vector space equipped with a nondegenerate symmetric bilinear form q ∈ Sym2V, identified in this paper with its associated quadratic form
We assume that X is G-invariant for the action of a Lie group G ⊂ SO(V )
We denote by g = TeG the Lie algebra of G, where e is the identity element, note that g ⊂ so(V )
Summary
Let V be a complex vector space equipped with a nondegenerate symmetric bilinear form q ∈ Sym2V , identified in this paper with its associated quadratic form. Let X ⊂ V be an algebraic variety defined over R, this includes the case when X is the cone over a projective variety defined over R. We assume that X is G-invariant for the action of a Lie group G ⊂ SO(V ). In many cases of interest X is H -invariant for a larger group H and we can take G = SO(V ) ∩ H , see Section 2 for the case of partially symmetric tensors. We denote by g = TeG the Lie algebra of G, where e is the identity element, note that g ⊂ so(V ).
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