Hypergeometric Equations and Weighted Projective Spaces

We compute the Hodge numbers of the polarised (pure) variation of Hodge structure V = grn−1 R f! Z of the Landau-Ginzburg model f : Y → C mirror-dual to a weighted projective space wPn in terms of a variant of Reid’s age function of the anticanonical cone over wPn. This implies, for instance, that wPn has canonical singularities if and only if hn−1,0 V = 1. We state a conjectural formula for the Hodge numbers of general hypergeometric variations. We show that a general fibre of the Landau-Ginzburg model is birational to a Calabi-Yau variety if and only if a general anticanonical section of wP is Calabi-Yau. We analyse the 104 weighted 3-spaces with canonical singularities, and show that a general anticanonical section is not a K3 surface exactly in those 9 cases where a generic fibre of the Landau-Ginzburg model is an elliptic surface of Kodaira dimension 1.