User:Nsoum/Sternberg convexity theorem

In mathematics, the Atiyah / Guillemin - Sternberg theorem, proved independently by Michael Atiyah and Victor Guillemin - Shlomo Sternberg, states that for a compact and connected Hamiltonian $\mathbb {T} ^{n}$-manifold the image of a moment map is a convex polytope. It also explicitly describes the vertices of the polytope. The AGS theorem is a generalization of the Schur-Horn theorem. This result was further extended by Frances Kirwan to the context of non-abelian compact Lie group actions which is known as Kirwan convexity theorem.

Statement of the theorem

Let $\mathbb {T} ^{n}$ be the $n$ - dimensional torus group and let $M$ be a compact and connected manifold with a Hamiltonian $\mathbb {T} ^{n}$ - action with moment map $\Phi :M\rightarrow {\mathfrak {t}}^{*}$, where ${\mathfrak {t}}$ is the Lie algebra of $\mathbb {T} ^{n}$. Then $\Phi (M)$ is the convex polytope generated by the set $\{\Phi (p):p\in M,t\cdot p=p\;\;\forall \;t\in \mathbb {T} ^{n}\}$. In other words, $\Phi (M)$ is the convex polytope generated by the image under $\Phi$ of the fixed - point set of the $\mathbb {T} ^{n}$ - action on $M$.