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  "body": "\n## Definition\n\nLet $f \\colon (\\C^n, 0) \\to (\\C, 0)$ be a [[germ|VZXF7RNFKS]] of a [[holomorphic function|0NAHXR1FRH]] with an isolated [[critical point|0YTCMFB4HC]] at $0$.\nLet $k$ be so large that $f$ is $k$-determined.\nLet $J^k$ be the space of [[$k$-jets|1QE0Z0756D]] at $0$ of germs vanishing at $0$.\nThe group of $k$-jets of germs of biholomorphic maps $(\\C^n, 0) \\to (\\C^n, 0)$ acts on $J^k$ by composition on the right.\nThe *modality* of $f$ is the least integer $m \\ge 0$ with this property.\nSome neighbourhood of $j^k f$ in $J^k$ is covered by finitely many families of orbits, each depending on at most $m$ parameters.\nA germ of modality $0$ is called *simple*, of modality $1$ *unimodal*, and of modality $2$ *bimodal*.\n\n## Remarks\n\nInformally, a family of orbits depending on $m$ parameters is the union of the orbits of the points of an $m$-dimensional smooth submanifold of $J^k$.\nThe modality does not depend on the choice of $k$, as long as $f$ is $k$-determined.\nExample: $f = x^3$ on $\\C$ is simple.\nAny small perturbation of the jet of $x^3$ in $J^3$ is $a_1 x + a_2 x^2 + a_3 x^3$ with $a_3 \\ne 0$.\nIf $a_1 \\ne 0$ it is right equivalent to $x$, a regular point.\nIf $a_1 = 0 \\ne a_2$ it is right equivalent to $x^2$, a Morse point.\nIf $a_1 = a_2 = 0$ it is right equivalent to $x^3$.\nNon-example: $f = x^4 + y^4$ on $\\C^2$ is not simple.\nFor the germs $x^4 + y^4 + a x^2 y^2$ with $a^2 \\ne 4$, the zero set is four distinct lines through $0$.\nTheir cross-ratio, taken up to the order of the lines, depends on $a$ and is invariant under right equivalence.\nSo infinitely many orbits meet every neighbourhood of $f$, and in fact the modality of $f$ is $1$.\n",
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