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  "body": "\n## Example\n\nLet $G$ be the complex [[Heisenberg group|ZZYCV5EX9J]] of matrices\n$$\\begin{pmatrix} 1 \u0026 x \u0026 z \\\\ 0 \u0026 1 \u0026 y \\\\ 0 \u0026 0 \u0026 1 \\end{pmatrix}, \\qquad x, y, z \\in \\C,$$\na complex Lie group isomorphic to $\\C^3$ as a complex manifold.\nLet $\\Gamma \\subset G$ be the subgroup of matrices with $x, y, z$ in the [[Gaussian integers|0K5W69AC0J]] $\\Z[i]$.\nThe *Iwasawa manifold* is the quotient $X = \\Gamma \\backslash G$ of $G$ by the left action of $\\Gamma$.\nIt is a compact [[complex manifold|JSGWAN2RJV]] of dimension $3$: $\\Gamma$ is a discrete cocompact subgroup, a [[lattice|BTYF5Y5PFY]] of $G$, and acts freely by holomorphic maps.\n\nCheck that $X$ is not [[Kähler|X60JG9AEQJ]].\nLeft multiplication by the matrix with entries $(a, b, c)$ sends $(x, y, z)$ to $(x + a, \\, y + b, \\, z + a y + c)$.\nSo the holomorphic $1$-forms $dx$, $dy$ and $\\theta = dz - x\\, dy$ are invariant: $d(z + ay + c) - (x + a)\\, d(y + b) = dz - x\\, dy$.\nThey descend to holomorphic $1$-forms on $X$.\nBut $d\\theta = -dx \\wedge dy \\ne 0$, so $\\theta$ is not [[closed|0ZV14XRNF0]].\nOn a compact Kähler manifold every holomorphic form is closed, so $X$ is not Kähler.\n\n## Remarks\n\n$X$ is a [[nilmanifold|43NRBE72C6]] and a fibre bundle over the complex torus $\\C^2 / \\Z[i]^2$, via $(x, y, z) \\mapsto (x, y)$, with fibre the elliptic curve $\\C / \\Z[i]$.\nIts holomorphic tangent bundle is trivial, spanned by the left-invariant vector fields dual to $dx, dy, \\theta$, so $X$ is a compact complex parallelizable manifold.\nIts first Betti number is $4$, an even number, so non-Kählerness is not visible from $b_1$ here.\n\n## Explanations\n\n### L1 human\n\nThe Iwasawa manifold is made from $3 \\times 3$ upper triangular complex matrices with ones on the diagonal.\nTwo matrices count as the same when they differ by one with Gaussian integer entries.\nIt is a compact complex manifold of dimension $3$ with three independent holomorphic vector fields at each point.\nIt has a holomorphic $1$-form whose derivative is not zero, which cannot happen on a compact Kähler manifold.\nSo it is a basic example of a compact complex manifold that is not Kähler, even though its Betti numbers look normal.\n",
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