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  "body": "\n## Definition\n\nLet $X$ be a compact [[complex manifold|JSGWAN2RJV]].\nA *complex analytic family* of compact complex manifolds is a [[holomorphic map|3MQTWKBBBT]] $\\varpi \\colon \\mathcal{X} \\to B$ between connected complex manifolds such that:\n\n1. $\\varpi$ is [[proper|15XEBGNGHC]];\n2. $\\varpi$ is a submersion: its differential is surjective at every point of $\\mathcal{X}$.\n\nThen each fibre $X_t = \\varpi^{-1}(t)$ is a compact complex manifold.\nA *deformation* of $X$ is such a family together with a point $0 \\in B$ and a [[biholomorphism|FX92PP3HZR]] $X \\to X_0$.\n\n## Remarks\n\nTwo deformations of $X$ over the same pointed base are isomorphic if there is a biholomorphism of the total spaces over $B$ compatible with the identifications of $X_0$ with $X$.\nBy Ehresmann's theorem, conditions 1 and 2 imply that $\\varpi$ is a smooth fiber bundle, so all fibres are diffeomorphic to $X$.\nThe fibres need not be biholomorphic to each other.\n\nExample: let $\\mathbb{H}$ be the upper half-plane and let $\\Z^2$ act on $\\C \\times \\mathbb{H}$ by $(m, n) \\cdot (z, \\tau) = (z + m + n\\tau, \\tau)$.\nThe quotient $\\mathcal{X}$ is a complex manifold, and the projection $\\mathcal{X} \\to \\mathbb{H}$ is a proper submersion with fibre $\\C / (\\Z + \\tau\\Z)$ over $\\tau$.\nSo it is a deformation of the elliptic curve $\\C/(\\Z + i\\Z)$ over $(\\mathbb{H}, i)$.\n\nNon-example: remove one point from the total space $\\mathcal{X}$ of the previous family.\nThe restricted projection is still a holomorphic submersion, but it is not proper, since one fibre is now a punctured torus, which is not compact.\n",
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