The local monodromy as a generalized algebraic correspondence
  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 $X \times X$. More precisely, they also define algebraic cycles on the special fibre of a resolution of $Z \times Z$. 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 $X \times X$. 
Caterina Consani