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 P\B, 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 2chains whose boundaries lie on Bφ, where {Bφ} 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 in a Calabi-Yau 3-fold ([CKYZ]). In the classification theory of algebraic varieties, one can study non-compact algebraic manifolds by means of Log Geometry. For a quasi-projective manifold U , there exist a projective varietyX and a normal crossing divisor B on X such that X\B is isomorphic to U . A rational differential form on X is called a logarithmic form if it can be locally written as a linear combination (with regular functions as coefficients) of products of d log f , where f is a regular function that vanishes at most in B. Then the space of logarithmic forms is an invariant of U and it plays the role that the space of regular forms plays in the classification theory of projective manifolds. In particular, we may think of a pair (X,B) (or the open variety X\B) as a log Calabi-Yau manifold if KX +B is trivial. It is expected that some analogue of Mirror Symmetry holds for log Calabi-Yau manifolds. As an evidence, we studied Mirror Symmetry for the one-dimensional log CalabiYau manifold, i.e. (P, {0,∞}) or C, in [T2]: for b ≥ 0 and k, l > 0, we consider covers of P that have k and l points over 0 and ∞ and are simply branched over b prescribed points. We define Fb,k,l(z1, . . . , zk;w1, . . . , wl) to be the generating function of the number of such curves. Then Fb,k,l coincides with the sum of certain integrals associated to graphs that have k and l ‘initial’ and ‘final’ vertices, have b internal vertices and are trivalent at internal vertices (here, variables z1, . . . , zk and w1, . . . wl are attached to initial and final vertices). The aim of this paper is to study the two-dimensional case as a further example of Log Mirror Symmetry. In Section 1, we describe Mirror Symmetry of P\B where B is a smooth cubic. On A-model side, we count curves of each degree in P\B whose normalizations are A and construct a power series from those numbers. On B-model side, we construct a function from the integrals of dx/x∧ dy/y over 2-chains whose boundaries lie on Bφ, where Bφ is a member of a family of smooth cubics parametrized by φ. Then we see that the coefficients of their expansions coincide up to order 8. Then, in Section 2, we discuss the relation between our log mirror symmetry and local mirror symmetry studied in [CKYZ]. We explain that the number of “affine lines” is in some sense the dual of local Gromov-Witten invariants of P in a Calabi-Yau 3-fold.