This must be a naive question, but I'm wondering about the definition of quasi-unipotent monodromy for general families, 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 $\Delta^{\ast}$, i.e. for a 1-parameter family. In this case we know that for a fibration $f: X \rightarrow \Delta$, with monodromy representation $\rho : \pi_{1}(\Delta^{\ast}) \rightarrow Aut(H^{i}(X_{0}))$, the image $\rho (T)$ of a generator $T$ of $\pi_{1}(\Delta^{\ast})$ is a quasi-unipotent matrix. What is the correct generalization of this to arbitrary families? For example, in the 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 that all of the matrices in the monodromy group (image of the monodromy representation) should be quasi-unipotent or does it suffice that the image of the generators be quasi-unipotent?
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3$\begingroup$ In higher dimensions, I think quasi-unipotent monodromy only makes sense in the case of normal crossings boundary. In this case the local fundamental group is abelian, hence if each the image of each generator is quasi-unipotent, so is the image of any other element. $\endgroup$– nafCommented Jul 21, 2012 at 12:58
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$\begingroup$ You mean if the discriminant is not NC, the notion of quasi-unipotency is not defined? Is there a reference which discusses the quasi-unupotency in details? anyway thank you very much for your anser. $\endgroup$– JackCommented Jul 21, 2012 at 14:32
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$\begingroup$ You should check out the answer to this: mathoverflow.net/questions/1912/… $\endgroup$– Igor RivinCommented Jul 21, 2012 at 17:55
1 Answer
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.
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$\begingroup$ Thank you very much for your beautiful answer. I had never known of this general definition of quasi-unipotency. I have some more questions: First of all what is the best reference which discusses this definition and it's consequences. Secondly, if this is the definition of "quasi-unipotency", then what would be the definition of "unipotency" in general? $\endgroup$– JackCommented Jul 22, 2012 at 6:46
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1$\begingroup$ Ah, I noticed I missed an adjective in the comment - I was defining what it means for $G$ to be quasi-unipotent. I edited the answer to reflect this correctly. The unipotency of $mon$ is defined in a similar manner - we say that $mon$ is unpotent, when $G$ is a connected unipotent algebraic group, i.e. when when it coincides with its unipotent radical. The references are numerous and the applications are usually Hodge theoretic. The quasi-unipotency of a local systen can is also very useful when we compute cohomology. $\endgroup$ Commented Jul 23, 2012 at 3:00
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1$\begingroup$ Two classical references are the paper "Periods of integrals on algebraic manifolds III", Publ. Math. IHES 38 (1970) by Griffiths and the paper "Variation of Hodge structure: the singularities of the period mapping" Invent. math. 22 (1973) by Schmid. They in particular explain Borel's proof of the quasi-unipotency theorem that I mentioned above. There are many other modern references. For instance, you may want to take a look at the excellent book "Period mappings and period domains" by Carlson, Mueller-Stach, and Peters. $\endgroup$ Commented Jul 23, 2012 at 3:07