Towards mirror symmetry for varieties of general type
Mirror symmetry for Calabi-Yau threefolds was first observed in [\textit{P. Candelas} et al., Nucl. Phys., B 359, No. 1, 21--74 (1991; Zbl 1098.32506)], which studied the particular case of the quintic and its mirror. It leads to a map between `mirror' manifolds \(X, \tilde{X}\) that exchanges their Hodge numbers \(h^{1, 1}, h^{2, 1}\). Since then, there have been many extensions to this picture of mirror symmetry, including (among many other developments), an understanding in terms of T-duality by \textit{A. Strominger} et al. [Nucl. Phys., B 479, No. 1--2, 243--259 (1996; Zbl 0896.14024)] and a categorical formulation (homological mirror symmetry) by \textit{M. Kontsevich} [in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)]. \textit{A. Givental} [Prog. Math. 160, 141--175 (1998; Zbl 0936.14031)] extended mirror symmetry to encompass Fano varieties and described their Landau-Ginzburg (LG) mirrors. This construction also applies to not necessarily Fano complete intersections in a toric variety, which have a LG mirror that is not necessarily of the same dimension as the original variety. The authors of this paper take steps towards placing the general mirror correspondence in a more geometric context in the spirit of Strominger-Yau-Zaslow, and Kontsevich. Given \(S^{\vee}\) a complete intersection in a toric variety (with no restriction on \(K_{S^{\vee}}\)), the authors provide a mirror dual of the same dimension as \(S^{\vee}\). In particular, they suggest that the LG dual \(S= \mathrm{Sing } w^{-1}(0)\) needs to be furnished with the sheaf of vanishing cycles \(\mathcal{F}_S = \phi_{w, 0}\mathbb{C}[1]\), which has natural support on \(S\). This approach necessitates a partial compactification of the LG mirror. They verify the characteristic mirror duality of Hodge numbers for their mirror pair: \(h^{p, q}(S^{\vee}) = h^{d-p, q}(S, \mathcal{F}_S)\). Many supporting results, review of appropriate background material, and a discussion of the relation to homological mirror symmetry round out this impressive paper.
scientific article