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].

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].

Reviewer: A.Quiros (MR 95k:14032)

##### 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 |