Asymptotic Hodge theory and quantum products

Abstract

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.

References and comments added. To appear in "Advances in Algebraic Geometry Motivated by Physics", Ed. E. Previatto, Contemporary Mathematics