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Jul 12, 2014 at 16:18 comment added Albuquerque Dear Misha Verbitsky, in my mind there was only one question and "Kähler form" is just the associated 2-form $\omega$ on Hermitian manifolds, not necessarily closed. Thanks for the answer. Regarding the classification you refer, the subclass of connections I am looking for is strictly smaller, I guess. And still interesting...
Jul 4, 2014 at 21:21 history answered Misha Verbitsky CC BY-SA 3.0