Hypergeometric equations and weighted projective spaces

By Alessio Corti, V. Golyshev. 2006

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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 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$\mathbb{V} = 1$. 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a5d556c076314e3a72e53a976986f0191c24ffa4