The symplectic structure of a toric conic transform

Suppose that a compact r-dimensional torus T-r acts in a holomorphic and Hamiltonian manner on polarized complex d-dimensional projective manifold M, with nowhere vanishing moment map Phi. Assuming that Phi is transverse to the ray through a given weight nu, associated to these data there is a complex (d - r+ 1)-dimensional polarized projective orbifold (M) over cap (v)(referred to as the v-th conic transformof M). Namely, (M) over cap (v) is a suitable quotient of the inverse image of the ray in the unit circle bundle of the polarization of M. With the aim to clarify the geometric significance of this construction, we consider the special case where M is toric, and show that (M) over cap (v) is itself a Kahler toric orbifold, whose (marked) moment polytope is obtained from the one of M by a certain 'transform' operation (depending on Phi and nu). (c) 2024 The Author(s). Published by Elsevier B.V.