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Let $G$ be a connected reductive group acting on a complex vector space $V$and projective space ${\mathbb P}V$. Let $x\in V$ and ${\cal H}\subseteq {\calG}$ be the Lie algebra of its stabilizer. Our objective is to understand points$[y]$, and their stabilizers which occur in the vicinity of $[x]$. We constructan explicit ${\cal G}$-action on a suitable neighbourhood of $x$, which we callthe local model at $x$. We show that Lie algebras of stabilizers of points inthe vicinity of $x$ are parameterized by subspaces of ${\cal H}$. When ${\calH}$ is reductive these are Lie subalgebras of ${\cal H}$. If the orbit of $x$is closed this also follows from Luna's theorem. Our construction involves amap connected to the local curvature form at $x$. We apply the local model toforms, when the form $g$ is obtained from the form $f$ as the leading term of aone parameter family acting on $f$. We show that there is a flattening ${\calK}_0$ of ${\cal K}$, the stabilizer of $f$ which sits as a subalgebra of ${\calH}$, the stabilizer $g$. We specialize to the case of forms $f$ whose$SL(X)$-orbits are affine, and the orbit of $g$ is of co-dimension $1$. We showthat (i) either ${\cal H}$ has a very simple structure, or (ii) conjugates ofthe elements of ${\cal K}$ also stabilize $g$ and the tangent of exit. Next, weapply this to the adjoint action. We show that for a general matrix $X$, thesignatures of nilpotent matrices in its projective orbit closure (underconjugation) are determined by the multiplicity data of the spectrum of $X$.Finally, we formulate the path problem of finding paths with specificproperties from $y$ to its limit points $x$ as an optimization problem usinglocal differential geometry. Our study is motivated by Geometric ComplexityTheory proposed by the second author and Ketan Mulmuley.

Geometric Complexity Theory -- Lie Algebraic Methods for Projective Limits of Stable Points