Toric generalized K\"ahler structures

Abstract

Given a compact symplectic toric manifold $(M,\omega, \mathbb{T})$, we identify a class $DGK_{\omega}^{\mathbb{T}}(M)$ of $\mathbb{T}$-invariant generalized K\"ahler structures for which a generalisation the Abreu-Guillemin theory of toric K\"ahler metrics holds. Specifically, elements of $DGK_{\omega}^{\mathbb{T}}(M)$ 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 K\"ahler structure on $M$ can be explicitly deformed to a non-K\"ahler element of $DGK_{\omega}^{\mathbb{T}}(M)$ by adding a small multiple of $C$. This constitutes an explicit realization of a recent unobstructedness theorem of R. Goto, where the choice of a matrix $C$ corresponds to choosing a holomorphic Poisson structure. Adapting methods from S. K. Donaldson, we compute the moment map for the action of $\mathrm{Ham}(M,\omega)$ on $DGK_{\omega}^{\mathbb{T}}(M)$. 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.