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On the local monodromy theorem. (Exposé I: Autour du théorème de monodrome locale.) (French) Zbl 0837.14013
Fontaine, Jean-Marc (ed.), Périodes $$p$$-adiques. Séminaire de Bures- sur-Yvette, France, 1988. Paris: Société Mathématique de France, Astérisque. 223, 9-57 (1994).
This paper describes some aspects of local monodromy theory that have inspired the constructions of O. Hyodo and K. Kato [in the same volume, Astérisque 223, 221-268 (1994) and of J.-M. Fontaine [ibid. 223, 321-347 (1994)] regarding semistable Galois representations. Let $$R$$ be a Henselian valuation ring with residue field $$k$$ and fraction field $$K$$ for which we choose an algebraic closure $$\overline K$$. Let $$p$$ be the characteristic exponent of $$k$$ and $$\ell$$ a prime different from $$p$$. Write $$G : = \text{Gal} (\overline K/K)$$ for the absolute Galois group and $$I$$ for the inertia subgroup of $$G$$. If $$\rho : G \to \text{GL} (V)$$ is a (continuous) $$\ell$$-adic representation of $$G$$ in a finite-dimensional $$\overline \mathbb{Q}_\ell$$-vector space, $$\rho$$ is said to be quasi-unipotent if there exists an open subgroup $$I_1$$ of $$I$$ such that the restriction of $$\rho$$ to $$I_1$$ is unipotent, that is, $$\rho - 1$$ is nilpotent. Whenever we have a quasi-nilpotent representation, there exists a unique nilpotent endomorphism $$N$$ of $$V$$, called the logarithm of the unipotent part of local monodromy, which in fact behaves as a logarithm of $$\rho$$ on $$I_1$$. The endomorphism $$N$$ allows one to define the local monodromy filtration on $$V$$. Grothendieck’s local monodromy theorem says that, if $$X$$ is a separated scheme of finite type over $$K$$ and $$H$$ is one of the cohomology groups $$H^n_c (X_{\overline K}, \overline \mathbb{Q}_\ell)$$ or $$H^n (X_{\overline K}, \overline \mathbb{Q}_\ell)$$ (which are finite- dimensional), the $$\ell$$-adic representation $$\rho : G \to \text{GL} (H)$$ coming from the action of $$G$$ on $$X_{\overline K}$$ is quasi- unipotent. After recalling these notions, in section 2 the author works over $$\mathbb{C}$$. Let $$f : X \to S$$ be a proper morphism of complex analytic spaces. If $$S$$ is an open disc and $$f$$ is smooth outside $$0 \in S$$, then the (positive) generator of $$\pi_1 (S - \{0\})$$ induces for each $$t \in S^* : = S - \{0\}$$ a local monodromy automorphism $$T$$ of $$H^*(X_t, \mathbb{Z})$$. The local monodromy theorem in this situation says that there exists an integer $$a$$ such that $$(T^a - 1)^{i + 1} = 0$$ on $$H^i (X_t, \mathbb{Z})$$ for all $$i$$. In case $$f$$ has semistable reduction at 0, this result may be explained in terms of the Gauss-Manin connection on the relative de Rham cohomology $$\mathbb{R}^* f_* \Omega^\bullet_{X^*/S^*}$$, where $$X^* : = f^{-1} (S^*)$$. More concretely, J. Steenbrink [Invent. Math. 31, 229-257 (1976; Zbl 0312.14007] tells us how to use the relative de Rham complex with logarithmic poles to calculate the complex of vanishing cycles and the logarithm $$N$$ of $$T$$. Moreover, Steenbrink’s theory provides us with a mixed Hodge structure, $$H^*_0$$, “limit as $$t \to 0$$” of the pure Hodge structures on $$H^* (X_t)$$. On $$H^*_0$$ one has two filtrations: the weight filtration of the mixed Hodge structure, and the monodromy filtration coming from $$N$$. That they coincide is probably the most profound result of the theory. All of these, as well as some consequences obtained by M. Saito, such as a generalization of the local invariant cycle theorem, are explained in section 2.
Section 3 treats some analogues of the previous theorems for $$S = \text{Spec} (R)$$ with residue characteristic $$p > 0$$ and $$f : X \to S$$ a proper morphism with semistable reduction. The author starts by explaining how Grothendieck, and later Rapoport and Zink, calculated the vanishing cycles, $$\mathbb{R}^i \Psi (\mathbb{Z}_\ell)$$, with the monodromy action, and in particular that $$I$$ acts trivially. This allows one to prove the local monodromy theory in this case. He also explains how M. Rapoport and T. Zink [Invent. Math. 68, 21-101 (1982; Zbl 0498.14010)] constructed an analogue of Steenbrink’s complex, thus getting a “weight spectral sequence” with abutment the cohomology of the geometric generic fiber. In this case the coincidence between the abutment filtration and the monodromy filtration is just the monodromy- weight conjecture, which is proved in several instances. The author ends by considering some questions of independence with respect to $$\ell$$. The perversity of the complex $$\mathbb{R}^i \Psi (\mathbb{Q}_\ell)$$ underlies the previous study. This rests on Artin’s theorem on the cohomological dimension of affine schemes and on the commutation of $$\mathbb{R} \Psi_\eta$$ with duality $$(\eta = \text{Spec} (K))$$. This last fact did not appear previously in the literature, and the paper ends by presenting a proof of it and some related items in section 4.
For the entire collection see [Zbl 0802.00019].

##### MSC:
 14F20 Étale and other Grothendieck topologies and (co)homologies 14G20 Local ground fields in algebraic geometry 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 32G05 Deformations of complex structures 32P05 Non-Archimedean analysis (should also be assigned at least one other classification number from Section 32-XX describing the type of problem) 14D07 Variation of Hodge structures (algebro-geometric aspects) 11G25 Varieties over finite and local fields