Log mirror symmetry and local mirror symmetry

摘要

We study Mirror Symmetry of log Calabi-Yau surfaces. On one hand, we consider the number of "affine lines" of each degree in P-2B, where B is a smooth cubic. On the other hand, we consider coefficients of a certain expansion of a function obtained from the integrals of dx/x Lambda dy/y over 2-chains whose boundaries lie on b phi, where {B-phi} is a family of smooth cubics. Then, for small degrees, they coincide. We discuss the relation between this phenomenon and local mirror symmetry for P-2 in a Calabi-Yau 3-fold ([CKYZ]). [References: 7]