Homological mirror symmetry in dimension one (Q2752196)

The celebrated homological mirror symmetry conjecture, formulated by \textit{M. Kontsevich} in his ICM talk in Zürich (1994), can be stated as follows:NEWLINENEWLINENEWLINEFor any Calabi-Yau threefold \(X\), there exist a symplectic mirror partner \(X^0\) and an equivalence of \((A_\infty)\)-categories \({\mathcal D}^b(X) \cong {\mathcal F}(X^0)\), where \({\mathcal D}^b(X)\) denotes the derived category of the abelian category of coherent sheaves on \(X\) and \({\mathcal F}(X^0)\) stands for the Fukaya \((A_\infty)\)-category associated with \(X^0\). Although formulated for Calabi-Yau threefolds, Kontsevich's homological mirror symmetry conjecture represents a great mathematical challenge in general, even so in the presumably simplest case of dimension one. The one-dimensional case, that is the case of an elliptic curve \(X\), has been investigated by \textit{A. Polishchuk} and \textit{E. Zaslow} in 1998 [Categorical mirror symmetry: The elliptic curve, Adv. Theor. Math. Phys. 2, 443-470 (1998; Zbl 0947.14017)]. These authors studied a somewhat weaker version of Kontsevich's conjecture, and their first attempt, in the simplest case, already demonstrated how difficult it might be to tackle Kontsevich's conjecture.NEWLINENEWLINENEWLINEThe paper under review grew out of the author's attempt to understand the details and subtleties in the paper of Polishchuk and Zaslow cited above. The result, and the main contribution of the present paper, is a correct formulation of the weakened version of Kontsevich's conjecture in dimension one and a now absolutely complete proof of it. Following the idea and strategy of the approach of Polishchuk-Zaslow, the author focuses on those parts of their paper that seem to require more explanation, rigor, or gapless justification, without repeating their entire proof of the homological mirror symmetry theorem in the one-dimensional case. In order to overcome the problem of dealing with Fukaya's \((A_\infty)\)-category, in this particular case, the author constructs directly a larger additive category \({\mathcal F}{\mathcal K}^0(X)\) and proves the homological mirror symmetry theorem for elliptic curves in the precise form \({\mathcal D}^b (X)\cong {\mathcal F}{\mathcal K}^0 (X^0)\).NEWLINENEWLINENEWLINEAltogether, the present paper provides a rewarding contribution towards establishing and better understanding the important results by Polishchuk and Zaslow.NEWLINENEWLINEFor the entire collection see [Zbl 0966.00024].