Date: Wednesday, February 23, 2022.
Time: 3:30 PM.
A Local Model Towards Understanding Projective Limits
Chennai Mathematical Institute.
Let G be a connected reductive algebraic group over C acting rationally on a complex vector space V and the corresponding projective space PV. Let x be a point in V, and let H be the stabilizer of x in, G. Our primary objective is to understand the points [y], and their stabilizers, which occur in the vicinity of [x] in PV. The motivation for studying this problem comes from algebraic complexity theory.
Towards this we construct an explicit action of Lie(G), the Lie algebra of G, on a suitably parametrized neighbourhood of x. As a consequence, we show that the Lie algebras of the stabilizers of points in the neighbourhood of x are parameterized by subspaces of Lie(H). When H is reductive, our results imply that the Lie algebras of points in the neighbourhood are in fact Lie subalgebras of Lie(H). If the orbit of x were closed this would also follow from a celebrated theorem of Luna. We call this data, of a neighbourhood of x with the Lie algebra morphism from Lie(G) to vector fields in this neighbourhood, a local model at x.
We illustrate the utility of the local model to understand when [x] is in the projective orbit closure of [y] via two applications. The applications we discuss are that of matrices under conjugation and limits of degree d forms in n variables.
This is joint work with Bharat Adsul and Milind Sohoni.