Toric generalized Kähler structures

Given a compact symplectic toric manifold $(M, \omega,\mathbb{T})$, we identify a class $DGK^{\mathbb{T}}_{\omega} (M)$ of $\mathbb{T}$-invariant generalized Kähler structures for which a generalisation the Abreu–Guillemin theory of toric Kähler metrics holds. Specifically, elements of $DGK^{\mathbb{T}}_{\omega} (M)$ are characterized by the data of a strictly convex function $\tau$ on the moment polytope associated to $(M, \omega, \mathbb{T})$ via the Delzant theorem, and an antisymmetric matrix $C$. For a given $C$, it is shown that a toric Kähler structure on $M$ can be explicitly deformed to a non-Kähler element of $DGK^{\mathbb{T}}_{\omega} (M)$ by adding a small multiple of $C$. This constitutes an explicit realization of a recent unobstructedness theorem of R. Goto [21, 22], where the choice of a matrix $C$ corresponds to choosing a holomorphic Poisson structure. Adapting methods from S. K. Donaldson [13], we compute the moment map for the action of $\mathrm{Ham} (M, \omega)$ on $DGK^{\mathbb{T}}_{\omega} (M)$. The result introduces a natural notion of “generalized Hermitian scalar curvature”. In dimension $4$, we find an expression for this generalized Hermitian scalar curvature in terms of the underlying bi-Hermitian structure in the sense of Apostolov–Gauduchon–Grantcharov [5].

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Received 1 March 2017

Accepted 18 August 2018

Published 24 October 2019