2018/11/14 by Yicao Wang, Wang, Yicao · 1 citation
Mathematics · #53D05 #53D18 #53D20 #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #math.DG #msc:53D05 #msc:53D18 #msc:53D20
paper · pdf · doi:10.48550/arxiv.1811.06848
This is a continuation of my recent work in arXiv:1810.08265v1. 44 pages, no figures
arxiv created 2018/11/14 · openalex publication_date 2018/11/14 · arxiv updated 2018/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Anti-diagonal toric generalized Kahler structures of symplectic type on a compact toric symplectic manifold were investigated in \citeWang2 . In this article, we consider general toric generalized Kahler structures of symplectic type, without requiring them to be anti-diagonal. Such a structure is characterized by a triple (τ, C, F) where τ is a strictly convex function defined in the interior of the moment polytope Δ and C, F are two constant anti-symmetric matrices. We prove that underlying each such a structure is a canonical toric Kahler structure I0 whose symplectic potential is given by this τ, and when C=0 the generalized complex structure \mathbbJ1 other than the symplectic one arises from an I0-holomorphic Poisson structure β in a novel way not mentioned in the literature before. Conversely, given a toric Kahler structure with symplectic potential τ and two anti-symmetric constant matrices C, F, the triple (τ, C, F) then determines a toric generalized Kahler structure of symplectic type canonically if F satisfies additionally a certain positive-definiteness condition. In particular, if the initial toric Kahler manifold is the standard MΔ associated to a Delzant polytope Δ, the resulting generalized Kahler structure can be interpreted as obtained via generalized Kahler reduction from a generalized Kahler structure on an open subset of a complex linear space, just as in Delzant's construction MΔ is obtained through Kahler reduction from a complex linear space.