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Let K be a field of characteristic zero that is complete with respect to a discrete valuation, and with perfect residue field. We formulate the notion of totally degenerate reduction for a smooth projective variety X over K. We show that for all prime numbers ℓ, the Q ℓ -étale cohomology of such a variety is (after passing to a finite unramified extension of K) a successive extension of direct sums of Galois modules of the form Q ℓ (r). More precisely, this cohomology has an increasing filtration whose r-th graded quotient is of the form V ⊗ Q Q ℓ (r), where V is a finite dimensional Q-vector space that is independent of ℓ, with an unramified action of the absolute Galois group of K.

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