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  "body": "\n## Summary\n\nAn almost complex manifold is a smooth manifold whose tangent spaces carry a smoothly varying multiplication by $i$.\nEvery complex manifold is one, but the converse fails in general.\nThe [[Newlander–Nirenberg theorem|VCT848WV1T]] says exactly when an almost complex structure comes from complex coordinates.\nAlmost complex structures are basic in Hermitian, Kähler and symplectic geometry.\n\n## Explanations\n\n### L1 human\n\nIn the complex plane, multiplying by $i$ turns each arrow by a quarter turn.\nAn almost complex structure gives such a quarter turn $J$ on every tangent plane of a manifold, with $J$ applied twice giving minus the identity.\nOn $\\R^2$ the rule $J(a, b) = (-b, a)$ is the quarter turn.\nOnly manifolds of even dimension can carry such a structure.\n",
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