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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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Yes, this is right. I don't think there's any deep reason why two integrable almost complex structures give rise to an integrable one, we just have to plough through the calculations to show it. If calculating the Nijenhuis tensor of $J_M \oplus J_N$ seems tiring, perhaps you could try showing the equivalent condition that $\bar \partial_{J_M \oplus J_N}^2 = 0$. – Gunnar Þór Magnússon Nov 1 '12 at 7:06
how could one check such a thing? – Markus Nov 1 '12 at 7:42
For any form $\alpha$ and almost complex structure $J$ we have (by definition) $$\bar \partial_J \alpha = \tfrac 12 (d\alpha \pm i d J \alpha),$$ where $d$ is the exterior derivative and I can never remember whether there should be a $+$ or $-$ in the place of $\pm$ (one gives $\partial$, the other gives $\bar \partial$). Integrability of $J$ (i.e. the vanishing of the Nijenhuis tensor) is equivalent to either $\partial^2 = 0$ or $\bar \partial^2 = 0$. – Gunnar Þór Magnússon Nov 1 '12 at 8:54
By the way, your question is independent of the Kahler condition. – Gunnar Þór Magnússon Nov 1 '12 at 8:56

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