A Note on Extremal Toric Almost Kähler Metrics

Abstract

An almost Kähler structure is extremal if the Hermitian scalar curvature is a Killing potential (Lejmi, Int J Math 21(12):1639–1662, 2010). When the almost complex structure is integrable it coincides with extremal Kähler metric in the sense of Calabi (Extremal Kähler metrics. II. In: Chavel I, Farkas HM (eds) Differential geometry and complex analysis. Springer, Berlin, 1985, pp 95–114). We observe that the existence of an extremal toric almost Kähler structure of involutive type implies uniform K-stability and we point out the existence of a formal solution of the Abreu equation for any angle along the invariant divisor. Applying the recent result of Chen and Cheng (On the constant scalar curvature Kähler metrics (III), General automorphism group. ArXiv1801.05907v1) and He (On Calabi’s extremal metric and properness. arXiv:math.DG/1801.07636), we conclude that the existence of a compatible extremal toric almost Kähler structure of involutive type on a compact symplectic toric manifold is equivalent to its relative uniform K–stability (in a toric sense). As an application, using (Apostolov et al., Adv Math 227:2385–2424, 2011), we get the existence of an extremal toric Kähler metric in each Kähler class of $\mathbb P(\mathcal {O}\oplus \mathcal {O}(k_1) \oplus \mathcal {O}(k_2))$.