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I have the following question: Let $(M,J_{M}, \omega_{M})$ and $(N,J_{N}, \omega_{N})$ be two Kähler manifolds. I consider $M \times N$. Can I introduce on $M \times N$ a complex structure ? I think one can define the complex structure by $J_{M} \oplus J_{N}$. Is this right? But then I do not understand why the Nijenhuis-tensor vanishes. Can someone explain this to me please. Thanks in advance.

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    By the way, your question is independent of the Kahler condition.2012-11-01

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