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  "body": "\n## Definition\n\nLet $n \\ge 0$ and let $M$ be a [[Hausdorff|0S1FJ8VEX9]], [[second-countable|03PPNCM9GG]] topological space.\nA *complex chart* on $M$ is a pair $(U, \\varphi)$ with $U \\subset M$ open and $\\varphi$ a [[homeomorphism|0QT00PB5G2]] of $U$ onto an open subset of $\\C^n$.\nA *holomorphic atlas* is a family of complex charts $(U_\\alpha, \\varphi_\\alpha)$ such that:\n\n1. the sets $U_\\alpha$ cover $M$;\n2. for all $\\alpha, \\beta$, the transition map $\\varphi_\\beta \\circ \\varphi_\\alpha^{-1} \\colon \\varphi_\\alpha(U_\\alpha \\cap U_\\beta) \\to \\varphi_\\beta(U_\\alpha \\cap U_\\beta)$ is [[holomorphic|3MQTWKBBBT]].\n\nTwo holomorphic atlases are equivalent if their union is a holomorphic atlas.\nA *complex manifold of dimension $n$* is such a space $M$ together with an equivalence class of holomorphic [[atlases|J7P32JQTVE]], called a *complex structure*.\nA function $f \\colon V \\to \\C$ on an open set $V \\subset M$ is holomorphic if $f \\circ \\varphi^{-1}$ is holomorphic for every chart $(U, \\varphi)$ of the structure.\n\n## Remarks\n\nExample: the [[Riemann sphere|0J78KBPFZB]] $\\hat\\C = \\C \\cup \\set{\\infty}$ with the charts $\\varphi_0(z) = z$ on $\\C$ and $\\varphi_\\infty(z) = 1/z$ on $\\hat\\C \\setminus \\set{0}$, with $\\varphi_\\infty(\\infty) = 0$.\nOn the overlap, $\\varphi_0(\\C \\setminus \\set{0}) = \\C \\setminus \\set{0}$, and the transition map $w \\mapsto 1/w$ is holomorphic there; so $\\hat\\C$ is a complex manifold of dimension $1$.\n\nNon-example: on $\\C$, the two charts $\\varphi_1(z) = z$ and $\\varphi_2(z) = \\bar z$ do not form a holomorphic atlas.\nEach is a homeomorphism onto $\\C$, but the transition map $w \\mapsto \\bar w$ is not holomorphic, so clause 2 fails.\n\nEvery complex manifold of dimension $n$ is a smooth manifold of dimension $2n$, since holomorphic maps are smooth.\nIt is canonically oriented, because the real Jacobian determinant of a holomorphic map is $\\abs{\\det J_\\C}^2 \\ge 0$, where $J_\\C$ is its complex Jacobian matrix.\nFurther examples are open subsets of $\\C^n$, [[complex projective space|7GJ4STCNNM]], [[complex tori|Y7MA6KBE9M]] and [[Hopf manifolds|Z8R2D33WSM]].\nA complex structure induces an [[almost complex structure|9SYWRGVTRS]]; by the [[Newlander–Nirenberg theorem|VCT848WV1T]], the almost complex structures that arise this way are exactly the integrable ones.\n",
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