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Hamiltonian systems, symplectic flows, classical integrable systems

1
vote
Consider the representation of $U(1)$ on $\mathbb{C}^n$ defined by $$t\cdot (z_1,\ldots,z_n)=(tz_1,\ldots,tz_n),$$ where $t\in U(1)$. This action is Hamiltonian.
answered Mar 13 '13 by Peter Crooks
5
votes
No, I think this need not be the case. Consider the usual action of $S^1$ on $\mathbb{C}^2$. The symplectic quotient is $\mathbb{P}^1$, which is not hyper-Kahler for dimension reasons.
answered Jun 2 '14 by Peter Crooks
1
vote
I am not offering an answer, but I suspect that the answer ultimately comes from the behaviour of Chern classes under pullbacks. More precisely, if $L\rightarrow M$ is an $S^1$-equivariant complex lin …
answered Apr 30 '14 by Peter Crooks
2
votes
I believe that the answer is yes. First note that the complexification $G_{\mathbb{C}}$ of $G$ is reductive and contains $G$ as a maximal compact subgroup. Secondly, $G_{\mathbb{C}}$ acts algebraica …
answered Nov 29 '14 by Peter Crooks
12
votes
2answers
Let $T$ be a compact real torus, and $X$ a Hamiltonian $T$-manifold (which you may take to be a smooth complex projective variety) with moment map $\mu:X\rightarrow\frak{t}^*$. If $\dim(T)=\frac{1}{2} …
asked Jul 27 '13 by Peter Crooks
3
votes
No, the bundle is not trivial in general. If we consider the case $n=1$, then the associated Lagrangian Grassmannian is actually the ordinary Grassmannian, namely $\mathbb{R}\mathbb{P}^1$. Also, the t …
answered Mar 2 '14 by Peter Crooks
5
votes
No, there are no counter-examples. Note that a generic coadjoint orbit is $G$-equivariantly diffeomorphic to $G/T$, for a maximal torus $T\subseteq G$. However, $G/T$ (also known as the full flag vari …
answered Feb 5 '14 by Peter Crooks
0
votes
I do not believe this is the case. If you have any smooth complex submanifold $X$ of $\mathbb{CP}^n$, then the Kahler form on $\mathbb{CP}^n$ pulls-back to a Kahler form $\omega$ on $X$ (so $\phi^*\om …
answered May 24 '14 by Peter Crooks
0
votes
Without knowing anything in particular about the $S^1$-action, your condition seems to me unlikely to be satisfied very often. Let $G=SL_2(\mathbb{C})$ and let $P$ be the standard Borel of upper-trian …
answered May 29 '14 by Peter Crooks
2
votes
1answer
Let $X$ be an affine algebraic variety over $\mathbb{C}$, and let $G$ be a semisimple complex linear algebraic group acting by variety automorphisms with finitely many orbits. The decomposition of $X$ …
asked Apr 27 '13 by Peter Crooks