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We classify compact self-dual almost-K\"ahler four manifolds of positive typeand zero type. In particular, using LeBrun's result, we show that any self-dualalmost-K\"ahler metric on a manifold which is diffeomorphic to$\mathbb{CP}_{2}$ is the Fubini-Study metric up to rescaling. In case ofnegative type, we classify compact self-dual almost-K\"ahler four manifoldswith $J$-invariant Ricci tensor.

Self-dual almost-K\"ahler four manifolds