The Picard-Lefschetz formula and a conjecture of Kato: The case of Lefschetz fibrations

The rational cohomology groups of complex algebraic varieties possess Hodge structures which can be used to refine their usual topological nature. Besides being useful parameters for geometric classification, these structures reflect in essential ways the conjectural category of motives, and hence, are of great interest from the viewpoint of arithmetic. A key link here is provided by the Hodge conjecture, which says that for smooth projective varieties, a class of type (n, n) in H should come from an algebraic cycle. Stated slightly more abstractly, a copy of the trivial Hodge structure inside H(X)(n) for a smooth projective variety X should be generated by an algebraic cycle class on X . Obviously, for the Hodge structures coming from smooth projective varieties, H(n) are the only ones that are of weight zero, and therefore, that can have trivial sub-structures. On the other hand, there are many important mixed Hodge structures (MHS) ([2] [3]) that arise from algebraic geometry, including open or singular varieties, local cohomologies, and the cohomology of Milnor fibers. The presence of trivial substructures is likely to reflect subtler phenomena (than the Hodge conjecture!) in the mixed case, having to do (possibly in a complicated way) with trivial substructures of various pure sub-quotients. The example we have in mind here is that of limit Hodge structures coming from degenerations ([5]). That is, let ∆ be the unit disk and X→∆