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Let $V$ be a real vector space. An almost complex structure on $V$ is a map $J : V \to V$ such that $J^2 = -\mathrm{id}_V$. An almost complex structure gives $V$ the structure of a complex vector space by defining $(a+bi)v = av + bJ(v)$. The idea behind this definition is that $J$ represents multiplication by $i$. What if you instead consider multiplication by other complex numbers?

Let $n > 2$ be a fixed positive integer and let $J : V \to V$ be such that $J^n = -\mathrm{id}_V$ which I will refer to as an alternative almost complex structure. One can regard $J$ as trying to capture multiplication by $\zeta = \exp\left(\frac{\pi i}{n}\right)$. In particular, as $\{1, \zeta\}$ is linearly independent over $\mathbb{R}$, it is a basis for $\mathbb{C}$ as a real vector space. One can then endow $V$ with a complex structure by defining $(a + b\zeta)v = av + bJ(v)$.

Do alternative almost complex structures give rise to the same results as the standard almost complex structures? In particular, in the case where we extend them to bundle endomorphisms of the tangent bundle of a manifold.

If not, what fails? If so, is dealing with almost complex structures easier than dealing with alternative almost complex structures?

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  • $\begingroup$ What results about almost complex structures (on manifolds) do you have in mind? The most interesting results I know happen when the almost complex structure is integrable. What should be the appropriate notion of an integrable alternative almost complex structure $J$? Should charts $(U, h)$ satisfy $dh \circ J = J_0 \circ dh$, where $J_0$ is an appropriately defined standard alternative almost complex structure on $\Bbb C^k$? $\endgroup$ – Henry T. Horton Mar 17 '13 at 18:12
  • $\begingroup$ @HenryT.Horton: I suppose integrability is the main concern, and your definition seems like a perfectly good one. I'd also be interested to see if there are any issues when combining an alternative almost complex structure with other structures such as a symplectic form or a Riemannian metric. $\endgroup$ – Michael Albanese Mar 19 '13 at 0:12
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I posted this question on MathOverflow and received an excellent answer there by Vladimir S Matveev. Anyone interested should check it out.

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