Log Mirror Symmetry and Local Mirror Symmetry

We study Mirror Symmetry of log Calabi–Yau surfaces. On one hand, we consider the number of “affine lines” of each degree in ℙ²B, 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∧dy/y over 2-chains whose boundaries lie on B _φ, where {B _φ} is a family of smooth cubics. Then, for small degrees, they coincide. We discuss the relation between this phenomenon and local mirror symmetry for ℙ² in a Calabi–Yau 3-fold ([CKYZ]). Received: 1 October 1999 / Accepted: 22 November 2000