# Search Results

Search type | Search syntax |
---|---|

Tags | [tag] |

Exact | "words here" |

Author |
user:1234 user:me (yours) |

Score |
score:3 (3+) score:0 (none) |

Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |

Views | views:250 |

Code | code:"if (foo != bar)" |

Sections |
title:apples body:"apples oranges" |

URL | url:"*.example.com" |

Bookmarks |
inbookmarks:mine inbookmarks:1234 |

Status |
closed:yes duplicate:no migrated:no wiki:no |

Types |
is:question is:answer |

Exclude |
-[tag] -apples |

For more details on advanced search visit our help page |

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

**2**answers

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

**1**answer

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