The symplectic structure of a toric conic transform

2022/05/21 by Roberto Paoletti, Paoletti, Roberto
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2205.10585

openalex publication_date 2022/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \boldsymbolν, associated to these data there is a complex (d-r+1)-dimensional polarized projective orbifold M_{\boldsymbolν} (referred to as the \boldsymbolν-th conic transform of M). Namely, M_{\boldsymbolν} 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_{\boldsymbolν} is itself a Kähler toric obifold, whose moment polytope is obtained from the one of M by a certain "transform", operation (depending on Φ and \boldsymbolν).

Related

• Toric generalized Kähler structures
• Remarks on asymptotic isometric embeddings of conic transforms for torus actions
• Picard group and quantization of toric orbifolds
• Concavity, Abel-transform and the Abel-inverse theorem in smooth complete toric varieties
• Kahler geometry of toric manifolds in symplectic coordinates