Kirwan surjectivity and Lefschetz-Sommese theorems for a generalized hyperkahler reduction

Let G be a compact Lie group. We study a class of Hamiltonian (G x S-1)-manifolds decorated with a function s with certain equivariance properties, under conditions on the G-action which we call of (semi-)linear type. In this context, a close analogue of hyperkahler reduction is defined, and our main result establishes surjectivity of an appropriate analogue of Kirwan's map. As a particular case, our setting includes a class of hyperkahler manifolds with trihamiltonian torus actions, to which our surjectivity result applies.