Timeline for connectedness of fibers of torus-equivariant moment maps
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| Oct 29, 2017 at 14:27 | comment | added | Qiao | Thanks a lot for the comments. I am interested in the case of a compact, connected Hamiltonian $T-$manifold $M$ with moment map $\mu: M \rightarrow \mathfrak{t}^*$. A theorem of Atiyah states that any non-empty preimage of $\mu$ is connected. This is closely related to the famous convexity result that $\mu(M)$ is convex in $\mathfrak{t}^*$. | |
| Oct 29, 2017 at 14:04 | vote | accept | Qiao | ||
| Oct 29, 2017 at 9:51 | answer | added | Friedrich Knop | timeline score: 2 | |
| Oct 25, 2017 at 23:12 | comment | added | user21574 | Example: For singular symplectic varieties in the sense of Beauville see Theorem 5.3 of numdam.org/article/AMBP_2006__13_2_209_0.pdf and use theorem 2.6, and Theorem 6.3 of arxiv.org/pdf/math/0112144.pdf and also p.8 of theorem of Mostow 1955 mat.ug.edu.pl/kwwk/2010/presentations/imykytyuk.pdf | |
| Oct 25, 2017 at 22:23 | comment | added | user21574 | Konp extended Kirwan theorem for Hamiltonian G-varieties, see Knop, Friedrich: A connectedness property of algebraic moment maps. J. Algebra 258 (2002), no. 1, 122–136. | |
| Oct 25, 2017 at 22:01 | comment | added | user21574 | Theorem of Kirwan :Let $M$ be a Hamiltonian $G$-manifold, if $M$ is connected and compact then the level sets of moment map are connected. Kirwan, F.: Convexity properties of the moment mapping III. Invent. math. 77 (1984), 547-552 | |
| Oct 25, 2017 at 21:52 | comment | added | user21574 | There are examples such that fibers of moment map may not be connected , for example non-compact symplectic toric manifold. See Atiyah M.F., Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1–15., and Guillemin V., Sternberg S., Convexity properties of the moment mapping, Invent. Math. 67 (1982), 491–513. | |
| Oct 25, 2017 at 21:38 | comment | added | user21574 | Example: Let $X\subset G/P$ be Schubert variety, then $\Phi:X\to Lie(T)^*$, Let $K^\mathbb C=G$, and $T$ denote a maximal torus in $K$, then fibers of this momentum map are connected subspaces of Schubert variety $X$ see arxiv.org/abs/math/0606474 | |
| Oct 25, 2017 at 21:09 | history | asked | Qiao | CC BY-SA 3.0 |