Spectral sequence of weights in Hyodo-Kato cohomology. (La suite spectrale des poids en cohomologie de Hyodo-Kato.) (French) Zbl 0834.14010

This very interesting paper treats a \(p\)-adic analogue of the following result of J. Steenbrink [Invent. Math. 31, 229-257 (1976; Zbl 0312.14007)] in complex Hodge theory. Let \(X\) be a complex analytic Kähler manifold, \(D\) the complex unit disk and \(f : X \to D\) a proper morphism of analytic spaces, smooth away from 0, and such that its fibre at the origin \(Y\) is a divisor with normal crossings whose components are smooth Kähler manifolds. The logarithm of the monodromy acts as a nilpotent endomorphism on the sheaf of limiting cycles \(R \Psi_f (\mathbb{C})\), and therefore induces on it a finite increasing filtration called the monodromy filtration. On the other hand, the cohomology of limiting cycles \(H^i (Y,R \Psi_f (\mathbb{C}))\) has a structure of mixed Hodge structure, whose weight filtration is the abutment filtration of the weight spectral sequence. Steenbrink proved that this spectral sequence degenerates at \(E_2\), and that the weight filtration on \(H^i (Y,R \Psi_f (\mathbb{C}))\) coincides with the monodromy filtration.
In the paper under review, the author constructs a Hyodo-Steenbrink complex that allows him to prove, in certain cases, an analogue of the above theorem for the cohomology of a proper and flat scheme with semistable reduction over a complete discrete valuation field. – Let \(K\) be such a field with ring of integers \(A\) and residue field \(\kappa\) perfect of characteristic \(p > 0\). Let \(W\) denote the ring of Witt vectors with coefficients in \(\kappa\), \(K_0\) the field of fractions of \(W\) and \(\sigma\) the Frobenius automorphism of \(W\). Let \(X\) be a proper and flat \(A\)-scheme with semistable reduction, and let \(Y\) be its special fibre which is supposed to be a sum of smooth divisors. To such an object O. Hyodo and K. Kato [“Semistable reduction and crystalline cohomology with logarithmic poles” in: Périodes \(p\)-adiques, Sémin. Bures sur-Yvette 1988, Exposé V, Astérisque 223, 221-268 (1994)] associated groups of “crystalline cohomology with logarithmic poles”, \(H^* (Y^\times, W^\times)\). These are \(W\)-modules of finite type together with a \(\sigma\)-semilinear isogeny \(\Phi\), called Frobenius, and a nilpotent linear monodromy operator \(N\). If \(X/A\) is smooth, \(H^* (Y^\times, W^\times)\) is the usual crystalline cohomology and the monodromy is 0. If \(K\) is of characteristic 0 there is a canonical isomorphism \(H^* (Y^\times, W^\times) \otimes K \simeq H^*_{\text{DR}} (X_K/K)\), where \(X_K\) is the generic fibre of \(X\).
Hyodo and Kato also proved that, as is the case for the usual crystalline cohomology, this crystalline cohomology for logarithmic schemes may be calculated by means of the de Rham-Witt complex \(W^\bullet_{\omega_Y}\). – The author modifies their construction (so that he does not need to assume that \(K\) is of unequal characteristic) and shows, as his key technical result (section 3), that \(W^\bullet_{\omega_Y}\), viewed as an object in the derived category \(D(Y_{\text{ét}}, W)\), underlies an object \((WA^\bullet, P_k)\), which he calls the complex of Hyodo-Steenbrink, in the filtered derived category, where \((P_k)\) is a finite increasing filtration, called the weight filtration, and \(WA^\bullet\) is endowed with a nilpotent endomorphism \(\nu\) lifting the monodromy on \(W^\bullet_{\omega_Y}\). This allows him to build a weight spectral sequence abutting to \(H^* (Y^\times, W^\times)\), analogous to that of Steenbrink. \(H^* (Y^\times, W^\times) \otimes K_0\) then has a weight filtration coming from this spectral sequence and also a monodromy filtration determined by \(N\). This is the \(p\)-adic analogue of the complex situation studied by Steenbrink, so that the author naturally arrives at conjecture 3.24: If \(Y\) is projective the weight spectral sequence degenerates at \(E_2\) modulo torsion. Conjecture 3.27: If \(Y\) is the reduction of a projective semistable \(X/A\) the weight filtration agrees with the monodromy filtration.
We must now assume, as the author does, that the residual field \(\kappa\) is finite. He then proves (theorem 3.32) that \((H^* (Y^\times, W^\times)\)/torsion, \(\Phi)\) is a mixed crystal in the sense of Faltings, and hence that conjecture 3.24 is true. He is also able to prove, after studying the duality pairing in this situation, that conjecture 3.27 holds for curves (theorem 5.3) and for surfaces (corollary 6.2.3).
As applications, he shows (proposition 5.9) that the conjecture of Fontaine, saying that an abelian variety \(A/K\) has good reduction if and only if the Galois representation on its Tate module is crystalline, holds if \(\kappa\) is finite and \(A\) is potentially a product of Jacobians. He also gives an explicit formula, in terms of the Rham cohomology, for the local factor at a prime of semistable reduction of the Hasse-Weil zeta function of a surface.


14F30 \(p\)-adic cohomology, crystalline cohomology
14K15 Arithmetic ground fields for abelian varieties
55T25 Generalized cohomology and spectral sequences in algebraic topology
14F40 de Rham cohomology and algebraic geometry
14G20 Local ground fields in algebraic geometry


Zbl 0312.14007
Full Text: DOI


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