Diffeomorphisms and families of Fourier–Mukai transforms in mirror symmetry

Assuming the standard framework of mirror symmetry, a conjecture is formulated describing how the diffeomorphism group of a Calabi–Yau manifold Y should act by families of Fourier–Mukai transforms over the complex moduli space of the mirror X. The conjecture generalizes a proposal of Kontsevich relating monodromy transformations and selfequivalences. Supporting evidence is given in the case of elliptic curves, lattice-polarized K3 surfaces and Calabi–Yau threefolds. A relation to the global Torelli problem is discussed.