zbMATH — the first resource for mathematics

Degeneration of \(l\)-adic weight spectral sequences. (English) Zbl 1033.14012
From the introduction: Let \(A\) be a henselian discrete valuation ring and let \(l\) be a prime number invertible in \(A\). We prove:
Theorem 1. For a proper semistable family over \(A\) whose special fiber has smooth irreducible components, the Steenbrink-Rapoport-Zink \(l\)-adic weight spectral sequence degenerates in \(E_2\)-terms.
The degeneracy was proved in cases where
(i) \(A=\mathbb{C}\{t\}\) by J. H. M. Steenbrink [Invent. Math. 31, 229–257 (1976; Zbl 0303.14002)]; and
(ii) The residue field of \(A\) is finite by a result by M. Rapoport and Th. Zink [Invent. Math. 68, 21-101 (1982; Zbl 0498.14010)].
The second case was proved by the Weil conjecture on Frobenius weights. L. Illusie [Astérisque 223, 9-57 (1994; Zbl 0837.14013)] conjectured that the degeneracy holds over an arbitrary A. Theorem 1 gives an affirmative answer to it.

14G20 Local ground fields in algebraic geometry
14F30 \(p\)-adic cohomology, crystalline cohomology
14D06 Fibrations, degenerations in algebraic geometry
Full Text: DOI Link