Take the 2-minute tour ×  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-unipote