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On the monodromy at isolated singularities of weighted homogeneous polynomials


Author: Benjamin G. Cooper
Journal: Trans. Amer. Math. Soc. 269 (1982), 149-166
MSC: Primary 32C40; Secondary 14B05
DOI: https://doi.org/10.1090/S0002-9947-1982-0637033-X
MathSciNet review: 637033
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Abstract: Assume $ f:{{\mathbf{C}}^m} \to {\mathbf{C}}$ is a weighted homogeneous polynomial with isolated singularity, and define $ \phi :{S^{2m - 1}} - {f^{ - 1}}(0) \to {S^1}$ by $ \phi (\overrightarrow z ) = f(\overrightarrow z ) / \vert f(\overrightarrow z )\vert$. If the monomials of $ f$ are algebraically independent, then the closure $ {\overline F _0}$ of $ {\phi ^{ - 1}}(1)$ in $ {S^{2m - 1}}$ admits a deformation into the subset $ G$ where each monomial of $ f$ has nonnegative real values. For the polynomial $ f({z_1}, \ldots ,{z_m}) = z_1^{{a_1}}{z_2} + \cdots + z_{m - 1}^{{a_{m - 1}}}{z_m} + z_m^{{a_m}}{z_1}$, $ G$ is a cell complex of dimension $ m - 1$, invariant under a characteristic map $ h$ of the fibration $ \phi $, and the inclusion $ G \to {F_0}$ induces isomorphisms in homology. To compute the homology of the link $ K = {f^{ - 1}}(0) \cap {S^{2m - 1}}$ it thus suffices to calculate the action of $ {h_{\ast}}$ on $ {H_{m - 1}}(G)$. Let $ d = {a_1}{a_2} \cdots {a_m} + {( - 1)^{m - 1}}$. Let $ {w_1},\,{w_2}, \ldots ,{w_m}$ be the weights associated with $ f$, satisfying $ {a_j} / {w_j} + 1 / {w_{j + 1}} = 1$ for $ j = 1,\,2, \ldots ,\,m - 1$ and $ {a_m}/{w_m} + 1/{w_1} = 1$. Let $ n = d/{w_1}$, $ q = \gcd (n,\,d)$, $ r = q + {( - 1)^m}$. Then $ {H_{m - 2}}(K) = {Z^r} \oplus {z_{d/q}}$ and $ {H_{m - 1}}(K) = {Z^r}$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1982-0637033-X
Keywords: Isolated singularity, weighted homogeneous polynomial, Milnor fibre, monodromy
Article copyright: © Copyright 1982 American Mathematical Society

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