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Abstract

The aim of this paper is to investigate the class of compact Hermitian surfaces $(M,g,J)$ admitting an action of the 2-torus $T^2$ by holomorphic isometries. We prove that if $b_1(M)$ is even and $(M,g,J)$ is locally conformally Kähler and $\chi (M)\not =0$ then there exists an open and dense subset $U\subset M$ such that $(U,g_{|U})$ is conformally equivalent to a 4-manifold which is almost Kähler in both orientations. We also prove that the class of Calabi Ricci flat Kähler metrics related with the real Monge–Ampère equation is a subclass of the class of Gibbons–Hawking Ricci flat self-dual metrics.