论文标题
$ U（2）$及其最大圆环的哈密顿动作的锥形减少
Conic reductions for Hamiltonian actions of $U(2)$ and its maximal torus
论文作者
论文摘要
假设在紧凑的Kähler歧管$ m $上，$ g = u（2）$的哈密顿和全态动作，无处消失的力矩图。如果在横向假设下，给出了$ g $的整体coadexhight轨道$ \ MATHCAL {O} $，我们将考虑两个自然相关的“圆锥”，减少。一个，将表示为$ \ overline {m}^g _ {\ mathcal {o}} $，是关于$ g $的动作和$ \ nathcal {o} $的圆锥的;另一个将表示为$ \叠加{m}^t _ {\boldsymbolν} $，是针对标准的最大圆环$ t \ leqslant g $和ray $ \ mathbb {r} _++++\，\ imath \ imath \ boldsymbolv $ cone cone o \ cone o \ cone o \ cone的行动。 Weyl腔室。这两个减少共享一个共同的“除数”，可以将其视为其结构之间的桥接。这一观点促使研究两种降低与后一个除数相关的（相当不同的）方式。在本文中，我们在此方向上提供了一些结果。此外，我们在投影设置中为大量此类动作提供明确的横向标准，并根据与动作和轨道相关的组合数据进行描述，以描述相应减少的加权投影品种。
Suppose given a Hamiltonian and holomorphic action of $G=U(2)$ on a compact Kähler manifold $M$, with nowhere vanishing moment map. Given an integral coadjoint orbit $\mathcal{O}$ for $G$, under transversality assumptions we shall consider two naturally associated 'conic', reductions. One, which will be denoted $\overline{M}^G_{\mathcal{O}}$, is taken with respect to the action of $G$ and the cone over $\mathcal{O}$; another, which will be denoted $\overline{M}^T_{\boldsymbolν}$, is taken with respect to the action of the standard maximal torus $T\leqslant G$ and the ray $\mathbb{R}_+\,\imath\boldsymbolν$ along which the cone over $\mathcal{O}$ intersects the positive Weyl chamber. These two reductions share a common 'divisor', which may be viewed heuristically as bridging between their structures. This point of view motivates studying the (rather different) ways in which the two reductions relate to the the latter divisor. In this paper we provide some results in this directions. Furthermore, we give explicit transversailty criteria for a large class of such actions in the projective setting, as well as a description of corresponding reductions as weighted projective varieties, depending on combinatoric data associated to the action and the orbit.
