Toric generalized Kähler structures (Q2336015)

Let \((M,\omega,\mathbb{T})\) be a compact symplectic toric manifold. The aim of this paper is to identify a natural notion of scalar curvature for a generalized Kähler structure, as defined and studied by \textit{M. Gualtieri} in [Generalized complex geometry. Oxford: Oxford University (PhD Thesis) (2004)] in the context of \textit{N. Hitchin}'s [Commun. Math. Phys. 265, No. 1, 131--164 (2006; Zbl 1110.53056)] generalized complex geometry. The approach used by the author to study this problem draws from the following three ingredients: the first concerns the interpretation of the scalar curvature as a moment map, the second ingredient is the computation by \textit{S. K. Donaldson} [J. Differ. Geom. 62, No. 2, 289--349 (2002; Zbl 1074.53059)] of this moment map in the context of the Abreu-Guillemin theory of toric Kähler metrics and the third ingredient is the notion of generalized Kähler structure of symplectic type and their realization as \(\omega\)-tamed complex structures. More precisely, the author identifies a class \(DGK^\mathbb{T}_\omega(M)\) of \(\mathbb{T}\)-invariant generalized Kähler structures for which a generalisation of 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 \textit{R. Goto} [J. Differ. Geom. 84, No. 3, 525--560 (2010; Zbl 1201.53085); J. Math. Soc. Japan 61, No. 1, 107--132 (2009; Zbl 1160.53014)], where the choice of a matrix \(C\) corresponds to choosing a holomorphic Poisson structure. Adapting methods from S. K. Donaldson, he computes the moment map for the action of \(Ham(M,\omega)\) on \(DGK^\mathbb{T}_\omega(M)\). The result introduces a natural notion of generalized Hermitian scalar curvature. In dimension \(4\), he finds an expression for this generalized Hermitian scalar curvature in terms of the underlying bi-Hermitian structure in the sense of \textit{V. Apostolov} et al. [Proc. Lond. Math. Soc. (3) 79, No. 2, 414--428 (1999; Zbl 1035.53061)].