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  "body": "\n## Example\n\nA *Hopf surface* is a compact complex surface whose [[universal cover|Y1YPE6PM52]] is biholomorphic to $\\C^2 \\setminus \\set{0}$.\nIt is *primary* if its fundamental group is infinite cyclic.\n\nExample: let $0 \u003c \\abs{\\lambda} \u003c 1$ and $X = (\\C^2 \\setminus \\set{0}) / \\langle g \\rangle$ with $g(z, w) = (\\lambda z, \\lambda w)$, the [[Hopf manifold|Z8R2D33WSM]] of dimension $2$.\nSince $\\C^2 \\setminus \\set{0}$ is homotopy equivalent to $S^3$, it is simply connected, so it is the universal cover of $X$, and $\\pi_1(X) \\cong \\Z$.\n\nCheck that $X$ is a [[class VII surface|Y4NSSJT4RC]].\nIts first [[Betti number|ST71ZV3FES]] is $1$, since $X$ is diffeomorphic to $S^1 \\times S^3$.\nIts [[Kodaira dimension|ZW00Q0SAGR]] is $-\\infty$.\nA section of $K_X^{\\otimes m}$ with $m \\ge 1$ pulls back to $h(z, w)\\,(dz \\wedge dw)^{\\otimes m}$ with $h$ holomorphic on $\\C^2 \\setminus \\set{0}$, hence, by Hartogs's extension theorem, on $\\C^2$.\nInvariance under $g$ gives $h(\\lambda z, \\lambda w)\\, \\lambda^{2m} = h(z, w)$, so each Taylor coefficient $c_{jk}$ of $h$ satisfies $c_{jk} \\lambda^{j + k + 2m} = c_{jk}$; since $\\abs{\\lambda} \u003c 1$, all $c_{jk}$ vanish and $h = 0$.\n\n## Remarks\n\nKodaira showed that every primary Hopf surface is the quotient of $\\C^2 \\setminus \\set{0}$ by the infinite cyclic group generated by a holomorphic contraction that fixes $0$.\nEvery Hopf surface has $b_1 = 1$ and $b_2 = 0$, so it is not Kähler.\nThe function $z/w$ on the example above is invariant under $g$, so it gives a nonconstant meromorphic function on $X$.\nIt also gives a map to $\\mathbb{CP}^1$ whose fibres are elliptic curves, and this $X$ has [[algebraic dimension|F0GF0JEZTG]] $1$.\n\n## Explanations\n\n### L1 human\n\nA Hopf surface is a compact complex surface obtained from $\\C^2$ minus the origin by gluing each point to its images under a contraction toward the origin.\nTopologically it is usually the product of a circle and a three-dimensional sphere.\nIt is the first and simplest example of a compact complex surface that is not Kähler, and so not algebraic.\n",
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