Diffeomorphisms and families of Fourier-Mukai transforms in mirror symmetry (Q2754426)

According with Kontsevich's proposal of homological mirror symmetry, there should be an equivalence between a category constructed from the bounded derived category of coherent sheaves on a Calabi-Yau and Fukaya's \(A_\infty\) category of special Lagrangian submanifolds of the mirror Calabi-Yau endowed with flat \(U(1)\)-bundles. Under this equivalence the monodromies of the 3-cycles should be mapped to certain automorphisms of the derived category, that is Fourier-Mukai transforms (Orlov). Kontsevich proposed the Fourier-Mukai transforms that should correspond to certain well-know monodromies and computed them explicitly for the quintic. Other explicit examples have been given by Andreas-Curio-Yau and the reviewer, by Horja and by \textit{S. Hosono} [Adv. Theor. Math. Phys. 4, No.~2, 335-376 (2000; Zbl 1008.14006)] in the toric case. NEWLINENEWLINENEWLINEIn the paper under review, families are introduced in the problem. Though families were implicit in the afore mentioned works, they are explicitly considered here for the first time in order to study the relationship between simplectomorphisms and monodromies. The author focusses in the determination of the action of the diffeomorphisms group of a Calabi-Yau manifold as a categorical mapping group of the complex moduli space of the mirror manifold. The precise formulation is stated as a conjecture. Evidences are nicely presented in the case of elliptic curves and of marked-polarized K3 surfaces. For three-dimensional Calabi-Yau only some examples are described. Very interesting is the example given in section 6.2 of a non-trivial action on the moduli spaces; the example is somehow inspired by earlier examples due to Horja, and it is achieved by combining birational contractions and Bondal-Orlov and Bridgeland results. The paper gives a series of counterexamples to the global Torelli for three dimensional Calabi-Yau manifolds, which are different in nature of the ones given by \textit{A. Cǎldǎraru} [J. Reine Angew. Math. 544, 161-173 (2002; Zbl 0995.14012)].NEWLINENEWLINENEWLINEThis paper is of expository nature, most of the proofs are postponed to future work.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00013].