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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $f$ be a holomorphic function near $0$ in $\\C^{n+1}$ with an [[isolated singularity|VTGYEES3VS]] at $0$.\nLet $F$ be the Milnor fibre of the [[Milnor fibration|KTFG1S2SMG]] of $f$ at $0$.\nGoing once around the base circle gives a diffeomorphism $h \\colon F \\to F$, defined up to isotopy, and its action $T = h_* \\colon H_n(F; \\C) \\to H_n(F; \\C)$ is the *[[monodromy|C8HD62AB6Q]]* of $f$.\nThen:\n\n1. every [[eigenvalue|0BQAA2BK3Q]] of $T$ is a [[root of unity|0YV2CKP2AV]];\n2. every [[Jordan block|07PH4RX0X5]] of $T$ has size at most $n + 1$.\n\nEquivalently, there is an integer $N \\ge 1$ such that $(T^N - \\id)^{n+1} = 0$: the monodromy is quasi-unipotent.\n\n## Remarks\n\nOne proof uses an embedded resolution of $f^{-1}(0)$, in which the pull-back of $f$ is locally a monomial $u \\, x_1^{N_1} \\cdots x_k^{N_k}$ with $u$ a unit.\nLocally the monodromy then has finite order, and one finds that $T^N$ is unipotent for $N$ the least common multiple of the multiplicities $N_i$.\nThe nilpotency index of $T^N - \\id$ is bounded by the number of divisors that can meet at a point, which is at most $n + 1$.\nOther proofs go through the regularity of the Gauss–Manin connection or through Hodge theory.\n\nExample: for $f = x^p + y^q$ with $p, q \\ge 2$, the monodromy on $H_1(F; \\C)$ is diagonalizable.\nIts eigenvalues are the products $\\zeta \\eta$, where $\\zeta \\ne 1$ is a $p$-th root of unity and $\\eta \\ne 1$ is a $q$-th root of unity.\nSo $T^N = \\id$ with $N = \\lcm(p, q)$.\n\n## Explanations\n\n### L1 human\n\nGoing once around the singular value, the Milnor fibre comes back to itself, but its cycles may be moved around.\nThe theorem says that this motion is almost periodic: after enough turns, it differs from doing nothing only by a nilpotent part.\nIn particular, no eigenvalue can be a number like $2$ or $e^{i\\sqrt{2}\\pi}$; they are all roots of unity.\n\n### L2 human\n\nThe monodromy theorem is the local version of a result for families of algebraic varieties over a punctured disc, proved in several ways by Grothendieck, Landman, Clemens and others.\nFor singularities, the eigenvalues of $T$ are linked to the roots of the [[Bernstein–Sato polynomial|7M5P2A0BZ6]] of $f$.\nBy results of Malgrange and Kashiwara, the numbers $e^{2\\pi i \\alpha}$ for the roots $\\alpha$ are the eigenvalues of the monodromy at points of $f^{-1}(0)$.\nThe size of the Jordan blocks is related to the weight filtration of the limit mixed Hodge structure on the cohomology of the Milnor fibre.\n",
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