In this paper we propose a process of lagrangian reduction and reconstruction for nonholonomic discrete mechanical systems where the action of a continuous symmetry group makes the configuration space a principal bundle. The result of the reduction process is a discrete dynamical system that we call the discrete reduced system. We illustrate the techniques by analyzing two types of discrete symmetric systems where it is possible to go further and obtain (forced) discrete mechanical systems that determine the dynamics of the discrete reduced system. 1991 Mathematics Subject Classification. Primary: 37J15, 37J60; Secondary: 70G75.

Assuming suitable convergence properties for the Gromov-Witten potential of a Calabi-Yau manifold X, one may construct a polarized variation of Hodge structure over the complexified Kähler cone of X. In this paper we show that, in the case of fourfolds, there is a correspondence between "quantum potentials" and polarized variations of Hodge structures that degenerate to a maximally unipotent boundary point. Under this correspondence, the WDVV equations are seen to be equivalent to the Griffiths' trasversality property of a variation of Hodge structure.

Abstract. We construct a polarized Hodge structure on the primitive part of Chen and Ruan's orbifold cohomology H k orb (X) for projective SL-orbifolds X satisfying a "Hard Lefschetz Condition". Furthermore, the total cohomology p,q H p,q orb (X) forms a mixed Hodge structure that is polarized by every element of the Kähler cone of X. Using results of Cattani-Kaplan-Schmid this implies the existence of an abstract polarized variation of Hodge structure on the complexified Kähler cone of X.This construction should be considered as the analogue of the abstract polarized variation of Hodge structure that can be attached to the singular cohomology of a crepant resolution of X, in light of the conjectural correspondence between the (quantum) orbifold cohomology and the (quantum) cohomology of a crepant resolution.

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