Compact quotients of complex homogeneous spaces with compact isotropy are Kähler whenever they admit both a balanced metric and a pluriclosed metric.
Arroyo and Marina Nicolini,SKT structures on nilmanifolds, Math
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The Fino--Vezzoni conjecture on homogeneous spaces
Compact quotients of complex homogeneous spaces with compact isotropy are Kähler whenever they admit both a balanced metric and a pluriclosed metric.