Log mirror symmetry and local mirror symmetry

Abstract

We study Mirror Symmetry of log Calabi-Yau surfaces. On one hand, we consider the number of ``affine lines'' of each degree in the complement of a smooth cubic in the projective plane. On the other hand, we consider coefficients of a certain expansion of a function obtained from the integrals of dxdy/xy over 2-chains whose boundaries lie on B_ϕwhere {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 projective plane in a Calabi-Yau 3-fold by Chiang-Klemm-Yau-Zaslow.

6 pages

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• A proof of N.Takahashi's conjecture for $({\mathbb{P}}^2,E)$ and a refined sheaves/Gromov-Witten correspondence
• Gromov-Witten Invariants of Local P^2 and Modular Forms
• ${\mathbb{A}}^1$ curves on log K3 surfaces
• Relative Gromov-Witten and maximal contact conics
• The local/logarithmic correspondence and the degeneration formula for quasimaps