The symplectic structure of a toric conic transform

TL;DR

This paper explores the geometric properties of a conic transform of a polarized complex manifold with torus action, showing that in the toric case, the transform results in a new Kähler toric orbifold with a modified moment polytope.

Contribution

It introduces the concept of a conic transform for polarized complex manifolds with torus actions and characterizes its structure in the toric case as a Kähler toric orbifold with a transformed moment polytope.

Findings

The conic transform produces a new polarized orbifold from the original manifold.

In the toric case, the transform yields a Kähler toric orbifold.

The moment polytope of the transformed manifold is explicitly related to the original polytope.

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