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Ricardo Andrade
Ricardo Andrade
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This must be a naive question, but I'm wondering about the definition of the quasi-unipotent monodromy for general (notfamilies, 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 ^{*}$$\Delta^{\ast}$, i.e. for a 1-parameter family. In this case we know that for a fibration $f: X \rightarrow \Delta$, with monodromy representation $\rho : \pi_{1}(\Delta^{*}) \rightarrow Aut(H^{i}(X_{0}))$$\rho : \pi_{1}(\Delta^{\ast}) \rightarrow Aut(H^{i}(X_{0}))$, the image $\rho (T)$ of a generator $T$ of $\pi_{1}(\Delta^{*})$$\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 quasiquasi-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 does it suffices onlysuffice that the image of the generators to be quasquasi-unipotent?

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?

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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Jack
Jack
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Quasi-unipotent monodromy for general families

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?