Hypergeometric equations and weighted projective spaces

Abstract

We compute the Hodge numbers of the polarised (pure) variation of Hodge structure $\mathbb{V} = gr_{n - 1}^W R^{n - 1} f_! \mathbb{Z}$ of the Landau-Ginzburg model f: Y → ℂ mirror-dual to a weighted projective space wℙ n in terms of a variant of Reid’s age function of the anticanonical cone over wℙ n . This implies, for instance, that wℙ n has canonical singularities if and only if h n−1,0 $\mathbb{V} = 1$ . 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.