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

Quasiunipotency 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 quasiunipotent 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 $mon : \pi_{1}(UD,o) \to GL(H^{i}(X_{o},\mathbb{C})$ has an image whose Zariski closure $G$ is a quasiunipotent linear algebraic group (that is, the quotient of $G$ by its unipotent radical is a finitegroup). In general it is rare for the local monodromy to be quasiunipotent. 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 quasiunipotent at all. However, if $p$ is at worst a normal crossing singularity of $D$, then the local monodromy is quasiunipotent. 

