The Picard-Lefschetz formula and a conjecture of Kato, the case of   Lefschetz fibrations
  A conjecture of Kato says that the monodromy operator on the cohomology of a semi-stable degeneration of projective varieties is represented by an algebraic cycle on the special fiber of a normal crossing model of the fiber product degeneration. We prove this conjecture in the simple case of a semi-stable degeneration arising from a Lefschetz fibration. 
Caterina Consani and Minhyong Kim