J functions, non-nef toric varieties and equivariant local mirror   symmetry of curves
  We develop techniques for computing the equivariant local mirror symmetry of curves, i.e. mirror symmetry for O(k)+O(-2-k) over P^1 for k greater than 0. We also describe related methods for dealing with mirror symmetry of non-nef toric varieties. The basic tools are equivariant I functions and their Birkhoff factorization. 
Brian Forbes (Hokkaido University), Masao Jinzenji (Hokkaido   University)