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Suppose given a Hamiltonian and holomorphic action of $G=U(2)$ on a compactK\"{a}hler manifold $M$, with nowhere vanishing moment map. Given an integralcoadjoint orbit $\mathcal{O}$ for $G$, under transversality assumptions weshall consider two naturally associated 'conic', reductions. One, which will bedenoted $\overline{M}^G_{\mathcal{O}}$, 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 thestandard 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 theirstructures. This point of view motivates studying the (rather different) waysin which the two reductions relate to the the latter divisor. In this paper weprovide some results in this directions. Furthermore, we give explicittransversailty criteria for a large class of such actions in the projectivesetting, as well as a description of corresponding reductions as weightedprojective varieties, depending on combinatoric data associated to the actionand the orbit.

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