On the monodromy at isolated singularities of weighted homogeneous polynomials

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 ${H_{m - 1}}(K) = {Z^r}$.