Equivalences of families of stacky toric Calabi-Yau hypersurfaces. (English) Zbl 1431.14040
The central results of this paper directly address the following question: given a pair of Fano toric varieties – and a pair of anti-canonical sections – can one provide natural conditions under which the zero loci of these sections define birational Calabi-Yau varieties? Alternatively, after a crepant partial resolution, when can we obtain a derived equivalence between such pairs of Calabi-Yau varieties?
As well as the general theory of toric varieties, the relevant aspects of which are summarised in the opening pages of the article, the central results draw on the derived equivalences obtained from variation of GIT by M. Ballard et al. [J. Reine Angew. Math. 746, 235–303 (2019; Zbl 1432.14015)] and D. Halpern-Leistner [J. Am. Math. Soc. 28, No. 3, 871–912 (2015; Zbl 1354.14029)], as well as the recent related work of D. Favero and T. L. Kelly [Adv. Math. 352, 943–980 (2019; Zbl 1444.14076)] on Berglund-Hübsch-Krawitz mirror symmetry.
The authors provide a wide array of applications for the derived and birational equivalences they obtain. In particlar, the authors describe birational and derived equivalences between mirrors constructed from anti-canonical hypersurfaces in the same toric Fano variety using a range of mirror constructions. These equivalences make use of the mirror construction of P. Clarke [Adv. Theor. Math. Phys. 21, No. 1, 243–287 (2017; Zbl 1386.81130)], which interprets various mirror constructions as an exchange of linear data associated to a pair of Landau-Ginzburg models via the sigma model/Landau-Ginzburg correspondence.
As well as the derived equivlance of Calabi-Yau mirrors, the authors obtain birational and derived equivalences between various \(K3\) surfaces obtained from the list of 95 weighted projective spaces whose anti-canonical linear system contain quasi-smooth \(K3\) surfaces with ADE singularities, recovering birational equivalences of M. Kobayashi and M. Mase [Tokyo J. Math. 35, No. 2, 461–467 (2012; Zbl 1262.14046)]. Moreover, applying their results to the case of quartic \(K3\) surfaces, the authors obtain equivalances between 52 of the 95 families in Reid’s list and linear systems of quartic surfaces.
As well as the general theory of toric varieties, the relevant aspects of which are summarised in the opening pages of the article, the central results draw on the derived equivalences obtained from variation of GIT by M. Ballard et al. [J. Reine Angew. Math. 746, 235–303 (2019; Zbl 1432.14015)] and D. Halpern-Leistner [J. Am. Math. Soc. 28, No. 3, 871–912 (2015; Zbl 1354.14029)], as well as the recent related work of D. Favero and T. L. Kelly [Adv. Math. 352, 943–980 (2019; Zbl 1444.14076)] on Berglund-Hübsch-Krawitz mirror symmetry.
The authors provide a wide array of applications for the derived and birational equivalences they obtain. In particlar, the authors describe birational and derived equivalences between mirrors constructed from anti-canonical hypersurfaces in the same toric Fano variety using a range of mirror constructions. These equivalences make use of the mirror construction of P. Clarke [Adv. Theor. Math. Phys. 21, No. 1, 243–287 (2017; Zbl 1386.81130)], which interprets various mirror constructions as an exchange of linear data associated to a pair of Landau-Ginzburg models via the sigma model/Landau-Ginzburg correspondence.
As well as the derived equivlance of Calabi-Yau mirrors, the authors obtain birational and derived equivalences between various \(K3\) surfaces obtained from the list of 95 weighted projective spaces whose anti-canonical linear system contain quasi-smooth \(K3\) surfaces with ADE singularities, recovering birational equivalences of M. Kobayashi and M. Mase [Tokyo J. Math. 35, No. 2, 461–467 (2012; Zbl 1262.14046)]. Moreover, applying their results to the case of quartic \(K3\) surfaces, the authors obtain equivalances between 52 of the 95 families in Reid’s list and linear systems of quartic surfaces.
Reviewer: Thomas Prince (Oxford)
MSC:
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 14C22 | Picard groups |
| 14J33 | Mirror symmetry (algebro-geometric aspects) |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 14J28 | \(K3\) surfaces and Enriques surfaces |
Keywords:
Calabi-Yau varieties; toric varieties; \(K3\) surfaces; derived equivalences; Picard groups; mirror symmetryReferences:
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