Abstract
We study compact complex manifolds bimeromorphic to locally conformally Kähler (LCK) manifolds. This is an analogy of studying a compact complex manifold bimeromorphic to a Kähler manifold. We give a negative answer for a question of Ornea, Verbitsky, Vuletescu by showing that there exists no LCK current on blow ups along a submanifold (dim \(\ge 1\)) of Vaisman manifolds. We show that a compact complex manifold with LCK currents satisfying a certain condition can be modified to an LCK manifold. Based on this fact, we define a compact complex manifold with a modification from an LCK manifold as a locally conformally class C (LC class C) manifold. We give examples of LC class C manifolds that are not LCK manifolds. Finally, we show that all LC class C manifolds are locally conformally balanced manifolds.
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Shimobe, H. Compact complex manifolds bimeromorphic to locally conformally Kähler manifolds. Geom Dedicata 197, 49–60 (2018). https://doi.org/10.1007/s10711-017-0317-y
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DOI: https://doi.org/10.1007/s10711-017-0317-y