Monodromy at Infinity and the Weights of Cohomology

We show that the size of the Jordan blocks with eigenvalue one of the monodromy at infinity is estimated in terms of the weights of the cohomology of the total space and a general fiber. Let f : X → S be a morphism of complex algebraic varieties with relative dimension n. Assume S is a smooth curve. Let U be a dense open subvariety of S such that the H(Xs,Q) for s ∈ U form a local system (which is naturally isomorphic to R f∗QX |U ), where Xs = f (s). It is known that the size of the Jordan blocks of the monodromy of H(Xs,Q) around any point of S \U (with arbitrary eigenvalues) does not exceed min{j+1, 2n− j+1}. See (2.4). However, it is often observed that the size of the Jordan blocks with eigenvalue one of the monodromy at infinity (i.e. around a point of S̄\S with S̄ the smooth compactification of S) is strictly smaller than the above estimate, e.g. if f : A → A is a polynomial map. See [5] in the case n = 1. In this paper, we show that this phenomenon is closely related to the vanishing of the cohomology of the total space, or more precisely, to the upper bound of the weights of the cohomology of a general fiber and the total space. 0.1. Theorem. Let f : X → S be a morphism of smooth complex algebraic varieties such that dimX = n+1 and dimS = 1. Let j, r be positive integers. Then the Jordan blocks of the monodromies at infinity of H(Xs,Q) with eigenvalue one have size ≤ r if H (X,Q) and H(Xs,Q) for a general s ∈ S have weights ≤ j + r. 0.2. Corollary. Assume X = A and S = A so that f is a polynomial map. Let j be a positive integer. Then the Jordan blocks with eigenvalue one of the monodromy at infinity of H(Xs,Q) have size ≤ j. Theorem mean that the restriction on the weights of the cohomology of the total space and a general fiber implies a certain restriction on the monodromy at infinity. (Note that the weights in the hypothesis are relevant to the mixed Hodge structure in [4], and not to the limit mixed Hodge structure.) The assumption on H(Xs,Q) is satisfied if Xs has a 1991 Mathematics Subject Classification. 32S40.