Duality of Mixed Hodge Structures of Algebraic Varieties

Abstract

which are dual to each other via Poincare pairings (cf. (1.5)). On the other hand, when Z is an algebraic variety (as we assume in the following), Deligne defined in [3] [4] the natural mixed (Q-)Hodge structure on each term of the above sequences, in such a way that the morphisms are those of mixed Hodge structures. The purpose of this article is then to show that the duality mentioned above is also compatible with the mixed Hodge structures under a suitable definition. A result in a sense analogous to ours has been obtained by Herrera and Lieberman in [13] in which they showed that the above duality is compatible with 'infinitesimal Hodge filtrations' of X along Y. Duality of mixed Hodge structure itself was also mentioned in the introduction of [4] as according to N. Katz. However, since there seems no published articles on this subject, it would not be of little use to give a detailed exposition like the present one. In Section 1 a precise statement of the theorem will be given and its proof is reduced to the case where we have to show that the pairing i^y: H*(Y9 Q)x Hy~(X, (?)->(? gives a duality of mixed Hodge structures under the assumption that Yis a divisor with only normal crossings in X. In this case we have