## On the monodromy at isolated singularities of weighted homogeneous polynomials

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- by Benjamin G. Cooper
- Trans. Amer. Math. Soc.
**269**(1982), 149-166 - DOI: https://doi.org/10.1090/S0002-9947-1982-0637033-X
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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 $\phi (\overrightarrow z ) = f(\overrightarrow z ) / |f(\overrightarrow z )|$. 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 ${H_{m - 1}}(K) = {Z^r}$.## References

- V. I. Arnol′d,
*Normal forms of functions in the neighborhood of degenerate critical points*, Uspehi Mat. Nauk**29**(1974), no. 2(176), 11–49 (Russian). Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I. MR**0516034** - Lawrence M. Graves,
*The theory of functions of real variables*, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1956. 2d ed. MR**0075256** - John Milnor,
*Singular points of complex hypersurfaces*, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR**0239612** - John Milnor and Peter Orlik,
*Isolated singularities defined by weighted homogeneous polynomials*, Topology**9**(1970), 385–393. MR**293680**, DOI 10.1016/0040-9383(70)90061-3 - Mutsuo Oka,
*On the homotopy types of hypersurfaces defined by weighted homogeneous polynomials*, Topology**12**(1973), 19–32. MR**309950**, DOI 10.1016/0040-9383(73)90019-0 - Peter Orlik,
*On the homology of weighted homogeneous manifolds*, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971) Lecture Notes in Math., Vol. 298, Springer, Berlin, 1972, pp. 260–269. MR**0430307** - Peter Orlik,
*Singularities and group actions*, Bull. Amer. Math. Soc. (N.S.)**1**(1979), no. 5, 703–720. MR**537624**, DOI 10.1090/S0273-0979-1979-14643-3 - P. Orlik and R. Randell,
*The monodromy of weighted homogeneous singularities*, Invent. Math.**39**(1977), no. 3, 199–211. MR**460320**, DOI 10.1007/BF01402973 - Peter Orlik and Philip Wagreich,
*Isolated singularities of algebraic surfaces with C$^{\ast }$ action*, Ann. of Math. (2)**93**(1971), 205–228. MR**284435**, DOI 10.2307/1970772
—,

*Algebraic surfaces with*${k^{\ast }}$-

*action*, Acta. Math.

**138**(1977), 43-81.

## Bibliographic Information

- © Copyright 1982 American Mathematical Society
- 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