A note on extremal toric almost K\"ahler metrics

Abstract

An almost K\"ahler structure is {\it extremal} if the Hermitian scalar curvature is a Killing potential [29]. When the almost complex structure is integrable it coincides with extremal K\"ahler metric in the sense of Calabi [8]. We observe that the existence of an extremal {\it toric} almost K\"ahler 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--Cheng [10] and He [27], we conclude that the existence of a compatible extremal toric almost K\"ahler 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 [5], we get the existence of an extremal toric K\"ahler metric in each K\"ahler class of $\mathbb{P}(\mathcal{O}\oplus \mathcal{O}(k_1) \oplus \mathcal{O}(k_2))$.