  We compute the Hodge numbers of the variation of Hodge structure of the
middle cohomology (with compact support) of the Landau-Ginzburg model dual to a
weighted projective space. We state a conjectural formula for the Hodge numbers
of general hypergeometric variations.
  We show that a general Landau-Ginzburg fibre is birational to a Calabi-Yau
variety if and only if a general anticanonical section of the weighted
projective space 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 Landau-Ginzburg fibre is an
elliptic surface of Kodaira dimension 1.
