Geodesic
Classification of non-K\"ahler surfaces and locally conformally K\"ahler geometry
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 76 (2021) no. 2, pp. 261-289

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The Enriques–Kodaira classification treats non-Kähler surfaces as a special case within the Kodaira framework. We prove the classification results for non-Kähler complex surfaces without relying on the machinery of the Enriques–Kodaira classification, and deduce the classification theorem for non-Kähler surfaces from the Buchdahl–Lamari theorem. We also prove that all non-Kähler surfaces which are not of class VII are locally conformally Kähler. Bibliography: 64 titles.
Keywords: locally conformally Kähler surface, Kato surface, elliptic fibration. 
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M. S. Verbitsky; V. Vuletescu; L. Ornea. Classification of non-K\"ahler surfaces and locally conformally K\"ahler geometry. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 76 (2021) no. 2, pp. 261-289. http://geodesic.mathdoc.fr/item/RM_2021_76_2_a1/