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Mar 14, 2011 at 21:47 vote accept Doris Hein
Mar 9, 2011 at 21:47 comment added Misha Verbitsky P. S. Of course, if you weaken your assumption by removing the Kaehler metric, the answer would be different - there are some examples of manifolds with big number of anticommuting complex structures, even compact ones, e.g. nilmanifolds.
Mar 9, 2011 at 8:27 history answered Misha Verbitsky CC BY-SA 2.5