The local monodromy as a generalized algebraic correspondence

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In the paper we show that for a normal-crossings degeneration Z over the ring of integers of a local field with X as generic fibre, the local monodromy operator and its powers determine invariant cocycle classes under the decomposition group in the cohomology of the product X × X. More precisely, they also define algebraic cycles on the special fibre of a resolution of Z × Z. In the paper, we give an explicit description of these cycles for a degeneration with at worst triple points as singularities. These cycles explain geometrically the presence of poles on specific local factors of the L-function related to X × X.