The symplectic structure of a toric conic transform

Abstract

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 Φ. Assuming that Φ is transverse to the ray through a given weight ν, associated to these data there is a complex $(d-r+1)$-dimensional polarized projective orbifold ${\hat{M}}_{\nu }$ (referred to as the ν-th conic transform of M). Namely, ${\hat{M}}_{\nu }$ 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 ${\hat{M}}_{\nu }$ is itself a Kähler toric orbifold, whose (marked) moment polytope is obtained from the one of M by a certain ‘transform’ operation (depending on Φ and ν).