Communications in Analysis and Geometry
Volume 13 (2005)
Mirror Symmetry Without Corrections
Pages: 287 – 331
We give geometric explanations and proofs of various mirror symmetry conjectures for Tn-invariant Calabi-Yau manifolds when instanton corrections are absent. This uses a fiberwise Fourier transformation together with a base Legendre transformation. We discuss mirror transformations of (i) moduli spaces of complex structures and complexified symplectic structures, Hp,q's, Yukawa couplings; (ii) sl (2) x sl (2)-actions; (iii) holomorphic and symplectic automorphisms and (iv) A- and B-connections, supersymmetric A- and B-cycles, correlation functions. We also study (ii) for Tn-invariant hyperkahler manifolds.