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A small clarification on bhargav's answer: in algebraic geometry we only have quasi-unipotency of the {\em local} monodromy in one parameter one-parameter families (which is what bhargav is talking about) about); or in multi-parameter families but only near a normal crossing point of the discriminant. Global monodromies are reductive and local monodromies near bad points of the discriminant can be more general.

For concreteness look at a projective morphism $f : X \to B$, where $X$, $B$ are smooth complex projective varieties. Let $D \subset B$ be the discriminant divisor of $f$, i.e. the divisor where the differential of $f$ is not surjective. The global monodromy of the smooth fibration $f : X - f^{-1}(D) \to B - D$ is always reductive by a theorem of Borel. That is: the Zariski closure of the monodromy in the linear automorphisms of the cohomology of the marked fiber is a complex reductive group. If we take a small analytic ball $U \subset B$ centered at some point of $D$, and if we know that $D\cap U$ is a normal crossings divisor in $U$, then the monodromy of the local fibration $f : f^{-1}(U-D) \to U-D$ is quasi-unipotent as bhargav explained. Note that the normal crossings condition implies that the fundamental group of $U - D$ is abelian, so the quasi-unipotency condition makes sense here.

If however $U\cap D$ does not have normal crossings, then $\pi_{1}(U-D)$ need not be abelian and the monodromy of $f : f^{-1}(U-D) \to U-D$ need not be quasi-unipotent. An easy example is to look at a generic projective plane $\mathbb{P}^{2}$ in the $9$ dimensional projective space of cubic curves in $\mathbb{P}^{2}$. This plane parametrizes a family of cubics which degenerates along a discriminant curve $D \subset \mathbb{P}^{2}$ and under the genericity assumption $D$ has only nodes and cusps. The cuspidal points of $D$ correspond to cuspidal cubics, and near a cusp of $D$ the local fundamental group of $U - D$ is the amalgamated product of $\mathbb{Z}/4$ and $\mathbb{Z}/6$ over $\mathbb{Z}/2$ and so is isomorphic to $SL_{2}(\mathbb{Z})$ the local monodromy representation near the cusp, i.e. the representation of $\pi_{1}(U-D,u_{0})$ to the linear automorphisms of the first integral cohomology of the cubic corresponding to $u_{0} \in U - D$, is an inclusion, i.e. has image $SL_{2}(\mathbb{Z})$. In particular it is not quasi-unipotent.

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A small clarification on bhargav's answer: in algebraic geometry we only have quasi-unipotency of the {\em local} monodromy in one parameter families (which is what bhargav is talking about) or in multi-parameter families but only near a normal crossing point of the discriminant. Global monodromies are reductive and local monodromies near bad points of the discriminant can be more general.

For concreteness look at a projective morphism $f : X \to B$, where $X$, $B$ are smooth complex projective varieties. Let $D \subset B$ be the discriminant divisor of $f$, i.e. the divisor where the differential of $f$ is not surjective. The global monodromy of the smooth fibration $f : X - f^{-1}(D) \to B - D$ is always reductive by a theorem of Borel. That is: the Zariski closure of the monodromy in the linear automorphisms of the cohomology of the marked fiber is a complex reductive group. If we take a small analytic ball $U \subset B$ centered at some point of $D$, and if we know that $D\cap U$ is a normal crossings divisor in $U$, then the monodromy of the local fibration $f : f^{-1}(U-D) \to U-D$ is quasi-unipotent as bhargav explained. Note that the normal crossings condition implies that the fundamental group of $U - D$ is abelian, so the quasi-unipotency condition makes sense here.

If however $U\cap D$ does not have normal crossings, then $\pi_{1}(U-D)$ need not be abelian and the monodromy of $f : f^{-1}(U-D) \to U-D$ need not be quasi-unipotent. An easy example is to look at a generic projective plane $\mathbb{P}^{2}$ in the $9$ dimensional projective space of cubic curves in $\mathbb{P}^{2}$. This plane parametrizes a family of cubics which degenerates along a discriminant curve $D \subset \mathbb{P}^{2}$ and under the genericity assumption $D$ has only nodes and cusps. The cuspidal points of $D$ correspond to cuspidal cubics, and near a cusp of $D$ the local fundamental group of $U - D$ is the amalgamated product of $\mathbb{Z}/4$ and $\mathbb{Z}/6$ over $\mathbb{Z}/2$ and so is isomorphic to $SL_{2}(\mathbb{Z})$ the local monodromy representation near the cusp, i.e. the representation of $\pi_{1}(U-D,u_{0})$ to the linear automorphisms of the first integral cohomology of the cubic corresponding to $u_{0} \in U - D$, is an inclusion, i.e. has image $SL_{2}(\mathbb{Z})$. In particular it is not quasi-unipotent.