Conic reductions for Hamiltonian actions of $U(2)$ and its maximal torus

Abstract: Suppose given a Hamiltonian and holomorphic action of $G=U(2)$ on a compact K\"{a}hler 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 $\overline{M}^T_{\boldsymbol{\nu}}$, is taken with respect to the action of the standard maximal torus $T\leqslant G$ and the ray $\mathbb{R}_+\,\imath\boldsymbol{\nu}$ 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.