Journal of Symplectic Geometry

Volume 21 (2023)

Number 2

Toric generalized Kähler structures. II

Pages: 235 – 264



Yicao Wang (Department of Mathematics, Hohai University, Nanjing, China)


Anti-diagonal toric generalized Kähler (GK) structures of symplectic type on a compact toric symplectic manifold were investigated in [$\href{}{18}$]. In this article, we consider general toric GK structures of symplectic type, without requiring them to be anti-diagonal. Such a structure is characterized by a triple $(\tau,C,F)$ where $\tau$ is a strictly convex function defined in the interior of the moment polytope $\Delta$ and $C, F$ are two constant anti-symmetric matrices. We prove that underlying each such a structure is a canonical toric Kähler structure $I_0$ whose symplectic potential is given by this $\tau$. Conversely, given a toric Kähler structure with symplectic potential $\tau$ and two anti-symmetric constant matrices $C, F$, the triple $(\tau,C,F)$ then determines a toric GK structure of symplectic type canonically if $F$ satisfies additionally a certain positive-definiteness condition.

This study is supported by the Natural Science Foundation of Jiangsu Province (BK20150797).

Received 7 February 2021

Received revised 26 January 2022

Accepted 15 August 2022

Published 28 September 2023