Mirror Symmetry of Fourier—Mukai Transformation for Elliptic Calabi—Yau Manifolds

Abstract

Mirror symmetry conjecture identi es the complex geometry of a CalabiYau manifold with the symplectic geometry of its mirror Calabi-Yau manifold. Using the SYZ mirror transform, we argue that (i) the mirror of an elliptic Calabi-Yau manifold admits a twin Lagrangian bration structure and (ii) the mirror of the Fourier-Mukai transform for dual elliptic brations is a symplectic Fourier-Mukai transform for dual twin Lagrangian brations, which is essentially an identity transformation in this case.