Self-dual almost-Kähler four-manifolds

We classify compact self-dual almost-K & auml;hler four-manifolds of positive type and zero type. In particular, using LeBrun's result, we show that any self-dual almost-K & auml;hler metric on a manifold which is diffeomorphic to CP2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {C}}}{{\mathbb {P}}}_{2}$$\end{document} is the Fubini-Study metric on CP2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {C}}}{{\mathbb {P}}}_{2}$$\end{document} up to rescaling. In case of negative type, we classify compact self-dual almost-K & auml;hler four-manifolds with J-invariant ricci tensor.