The Local Monodromy as a Generalized Algebraic Corresponcence

Let X be a proper and smooth variety over a local field K and let X be a regular model of X defined over the ring of integers OK of K. When X is smooth over OK , the Tate conjecture equates the l–adic Chow groups of algebraic cycles on the geometric special fibre Xk̄ of X → Spec(OK) with the Galois invariants in H(XK̄ ,Ql(∗)). One of the results proved in [2] (cf. Corollary 3.6) shows that the Tate conjecture for smooth and proper varieties over finite fields together with the monodromy–weight conjecture imply a generalization of the above result in the case of semistable reduction. Namely, let ℘ ∈ Spec(OK) be a place over which the special fibre X × Spec(k(℘)) = Y is a reduced divisor with normal crossings in X (i.e. semistable place). Then, assuming the above two conjectures, the l–adic groups of algebraic cycles modulo rational equivalence on the r–fold intersections of components of Y (r ≥ 1) are related with Galois invariant classes on the Tate twists H(XK̄ ,Ql(∗ − (r − 1))). An interesting case is when one replaces X by X ×K X, so that Galois invariant cycles may be identified with Galois equivariant maps H(XK̄ ,Ql) → H(XK̄ ,Ql(·)). Examples of such maps are the powers N i of the logarithm of the local monodromy around ℘. The operators N i : H(XK̄ ,Ql) → H(XK̄ ,Ql(−i)) determine classes [N ] ∈ H((X × X)K̄ ,Ql(d − i)) (d = dim XK̄) invariant under the decomposition group. In this paper we study in detail the structure of [N ] when the special fibre Y of X has at worst triple points as singularities. That is, we exhibit the corresponding algebraic cycles on the (normal crossings) special fibre T = ∪iTi of a resolution Z of X ×OK X . Denote by Ñ = 1 ⊗ N + N ⊗ 1 the monodromy on the product, and let F be the geometric Frobenius. Then the classes [N ] naturally determine elements in Ker(Ñ) ∩ H((X × X)K̄ ,Ql(d − i)). Assuming the monodromy–weight conjecture on the product (i.e. the monodromy filtration L · onH((X×X)K̄ ,Ql) coincides–up to a shift– with the filtration by the weights of the Frobenius cf. [14]), the following identifications hold