Monodromy theorem for singularities

Remarks

One 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. Locally the monodromy then has finite order, and one finds that $T^N$ is unipotent for $N$ the least common multiple of the multiplicities $N_i$. The nilpotency index of $T^N - \id$ is bounded by the number of divisors that can meet at a point, which is at most $n + 1$. Other proofs go through the regularity of the Gauss–Manin connection or through Hodge theory.

Example: for $f = x^p + y^q$ with $p, q \ge 2$, the monodromy on $H_1(F; \C)$ is diagonalizable. Its 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. So $T^N = \id$ with $N = \lcm(p, q)$.

Going deeper

For a student

The 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. For singularities, the eigenvalues of $T$ are linked to the roots of the Bernstein–Sato polynomial of $f$. By 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 $f^{-1}(0)$. The size of the Jordan blocks is related to the weight filtration of the limit mixed Hodge structure on the cohomology of the Milnor fibre.