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  "body": "\n## Example\n\nLet $n \\ge 2$ and let $\\lambda \\in \\C$ with $0 \u003c \\abs{\\lambda} \u003c 1$.\nLet $W = \\C^n \\setminus \\set{0}$ and let $\\Gamma$ be the group of maps $g_k(z) = \\lambda^k z$, $k \\in \\Z$.\nThe quotient $X = W / \\Gamma$ is a (classical) *Hopf manifold*, a compact [[complex manifold|JSGWAN2RJV]] of dimension $n$.\n\nCheck that $X$ is a complex manifold.\nThe action is free: if $\\lambda^k z = z$ with $z \\ne 0$, then $\\lambda^k = 1$, so $k = 0$.\nIt is [[properly discontinuous|J5X5ED0B2Y]]: a compact $K \\subset W$ lies in a shell $\\set{r \\le \\abs{z} \\le R}$ with $r \u003e 0$.\nThe set $g_k(K)$ lies in $\\set{\\abs{\\lambda}^k r \\le \\abs{z} \\le \\abs{\\lambda}^k R}$, which meets $K$ for only finitely many $k$.\nThe maps $g_k$ are holomorphic, so the [[quotient|0BR3FY5Q8M]] inherits a complex structure.\nIt is compact: it is the image of the compact shell $\\set{\\abs{\\lambda} \\le \\abs{z} \\le 1}$.\n\nFor $\\lambda$ real, the map $z \\mapsto (\\abs{z}, z/\\abs{z})$ identifies $W$ with $(0, \\infty) \\times S^{2n-1}$, and $\\Gamma$ acts only on the first factor.\nSo $X$ is diffeomorphic to $S^1 \\times S^{2n-1}$; the same holds for all $\\lambda$.\n\n## Remarks\n\n$X$ is not [[Kähler|X60JG9AEQJ]].\nOn a compact Kähler manifold of dimension $n$ the class of the Kähler form $\\omega$ in $H^2(X; \\R)$ is nonzero, since $\\int_X \\omega^n \u003e 0$.\nBut by the Künneth formula $H^2(S^1 \\times S^{2n-1}; \\R) = 0$ for $n \\ge 2$, because $2n - 1 \\ge 3$.\nAlso the first [[Betti number|ST71ZV3FES]] of $X$ is $1$, an odd number, which is impossible for a compact Kähler manifold.\n\nFor $n = 1$ the same construction gives $\\C^* / \\lambda^\\Z$, an elliptic curve, which is Kähler.\nFor $n = 2$ one gets the primary [[Hopf surfaces|PNFNGBXRM3]] of diagonal type.\n\n## Explanations\n\n### L1 human\n\nRemove the origin from $\\C^n$, and declare two points the same when one is $\\lambda$ times the other.\nWhat is left is a thick spherical shell whose inner and outer spheres are glued, which is the product of a circle and a sphere.\nThis simple space is a compact complex manifold that admits no Kähler metric.\nSo compact complex geometry is much wider than Kähler geometry, and hence than projective algebraic geometry.\n",
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