Question

This must be a naive question, but I'm wondering about the definition of the quasi-unipotent monodromy for general (not only 1-parameter families). The problem is that usually in the books of algebraic geometry quasi-unipotent monodromy is only discussed over a disc $\Delta ^{*}$ i.e. for a 1-parameter family. In this case we know that for a fibration $f: X \rightarrow \Delta$, with monodromy representation $\rho : \pi_{1}(\Delta^{*}) \rightarrow Aut(H^{i}(X_{0}))$, the image $\rho (T)$ of a generator $T$ of $\pi_{1}(\Delta^{*})$ is a quasi-unipotent matrix. What is the correct generalization of this to arbitrary families? For example, in multi-parameter case, it could happen that there are several generators $T_{i}$ such that each $\rho(T_{i})$ is a quasi-unipotent matrix but for example $\rho(T_{1})\rho(T_{2})$ is not quasi-unipotent . So is the possible generalization is that all of the matrices in the monodromy group (image of the monodromy representation) should be quasi-unipotent or it suffices only that the image of the generators to be quas-unipotent?

Answer 1

Quasi-unipotency is a well defined notion at any point of the discriminant. If we have a proper family $f : X \to S$ of varieties with a smooth total space and a smooth base, and if $p \in D \subset S$ is a point of the discriminant, then we say that the local monodromy of the family near $p$ is quasi-unipotent if we can find a small analytic neighborhood $p \in U \subset S$ of $p$ in $S$, so that if $o \in U - D$ is a base point, then the monodromy representation $mon : \pi_{1}(U-D,o) \to GL(H^{i}(X_{o},\mathbb{C})$ has an image whose Zariski closure $G$ is a quasi-unipotent linear algebraic group (that is, the quotient of $G$ by its unipotent radical is a finite-group).

In general it is rare for the local monodromy to be quasi-unipotent. If $p$ happens to be a very singular point of the discriminant, then the local monodromy tends to be big and is often as big as it can be, and not quasi-unipotent at all. However, if $p$ is at worst a normal crossing singularity of $D$, then the local monodromy is quasi-unipotent.