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  "body": "\n## Definition\n\nLet $U$ be an open subset of $\\C^{n+1}$ and let $f \\colon U \\to \\C$ be a [[holomorphic function|507C34W0ZT]].\nLet $p \\in U$ be a nondegenerate [[critical point|0YTCMFB4HC]] of $f$ with $f(p) = 0$.\nThis means that the partial derivatives of $f$ vanish at $p$ and the Hessian matrix $\\left( \\partial^2 f / \\partial z_j \\partial z_k (p) \\right)$ is invertible.\nBy the holomorphic Morse lemma there are holomorphic coordinates $w_0, \\dots, w_n$ centred at $p$, on a ball $B$ about $p$, in which $f = w_0^2 + \\cdots + w_n^2$.\nFor small real $t \u003e 0$, the set\n$$\\Sigma_t = \\set{w \\in B : w_0, \\dots, w_n \\in \\R, \\ w_0^2 + \\cdots + w_n^2 = t}$$\nis an $n$-dimensional [[sphere|P9CM4GQ2T0]] in the fibre $f^{-1}(t) \\cap B$.\nThe *vanishing cycle* at $p$ is the class $\\delta = [\\Sigma_t]$ in the [[homology group|M0Q5CFJ7ZH]] $H_n(f^{-1}(t) \\cap B; \\Z)$, defined up to the sign fixed by an orientation of $\\Sigma_t$.\n\n## Remarks\n\nWhen $B$ is a ball in the coordinates $w$, the class $\\delta$ generates the reduced homology $\\tilde H_n(f^{-1}(t) \\cap B; \\Z) \\cong \\Z$, and $\\Sigma_t$ shrinks to the point $p$ as $t \\to 0$.\nLet $f^{-1}(c)$ be a fibre further away, joined to $t$ by a path $\\gamma$ of regular values.\nTransporting $\\delta$ along $\\gamma$ gives a vanishing cycle in $H_n(f^{-1}(c); \\Z)$, which depends on the homotopy class of $\\gamma$.\n\nExample: $n = 0$ and $f(w) = w^2$.\nThe fibre $f^{-1}(t)$ is the two points $\\pm\\sqrt{t}$, and $\\delta = [\\sqrt{t}] - [-\\sqrt{t}]$ in $H_0$.\n\nExample: $n = 1$ and $f(w) = w_0^2 + w_1^2$.\nThe local fibre is an annulus, and $\\delta$ is the class of its core circle.\n\nNon-example: the critical point $0$ of $f(w) = w^3$ is degenerate, since the Hessian is $0$.\nIts local fibre is three points, with reduced $H_0$ of rank $2$, so it is not described by one vanishing cycle.\nAfter a small perturbation $w^3 - sw$ with $s \\ne 0$, it splits into two nondegenerate critical points, each with its own vanishing cycle.\n",
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