Mirror constructions for K3 surfaces from bimodal singularities. (English) Zbl 07919520
Abdellatif, Ramla (ed.) et al., Women in numbers Europe IV. Research directions in number theory. Selected papers based on the presentations at the 4th workshop, WINE 4, Utrecht, the Netherlands, August 29 – September 2, 2022. Cham: Springer. Assoc. Women Math. Ser. 32, 323-352 (2024).
Summary: Given a K3 surface realized as a hypersurface in a weighted projective space or a Gorenstein Fano toric variety, one may construct a mirror K3 surface in various ways. Depending on the precise model, available descriptions of mirror symmetry include the Greene-Plesser mirror, the Berglund-Hübsch transpose construction for invertible polynomials, Dolgachev-Nikulin-Pinkham’s lattice-polarized K3 surfaces, and Batyrev’s reflexive polytope construction. The multitude of descriptions raises the question of whether mirror constructions are consistent. Comparing different mirror constructions often entails making choices – one might need to specify a family containing a K3 surface or a lattice polarization, for example – and thus it is important to establish systematic methods for making these choices.
For the entire collection see [Zbl 1542.11001].
For the entire collection see [Zbl 1542.11001].
MSC:
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 14J28 | \(K3\) surfaces and Enriques surfaces |
| 14J33 | Mirror symmetry (algebro-geometric aspects) |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
Software:
SageMathReferences:
| [1] | M. Artebani, S. Boissière, A. Sarti, The Berglund-Hübsch-Chiodo-Ruan mirror symmetry for K3 surfaces. J. Math. Pures Appl. (9) 102(4), 758-781 (2014) · Zbl 1315.14051 |
| [2] | P.S. Aspinwall, B.R. Greene, D.R. Morrison, The monomial-divisor mirror map. Int. Math. Res. Notices 12, 319-337 (1993) · Zbl 0798.14030 |
| [3] | V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Alg. Geom. 3, 493-535 (1994) · Zbl 0829.14023 |
| [4] | U. Bruzzo, A. Grassi, Picard group of hypersurfaces in toric 3-folds. Int. J. Math. 23(2), 1-14 (2012) · Zbl 1252.14031 |
| [5] | P. Clarke, Duality for toric Landau-Ginzburg models. Adv. Theor. Math. Phys. 21(1), 243-287 (2017) · Zbl 1386.81130 |
| [6] | D.A. Cox, S. Katz, Mirror Symmetry and Algebraic Geometry (American Mathematical Society, Providence, RI, 1999) · Zbl 0951.14026 |
| [7] | I.V. Dolgachev, Mirror symmetry for lattice polarized \(K3\) surfaces. J. Math. Sci. 81(3), 2599-2630 (1996) · Zbl 0890.14024 |
| [8] | C.F. Doran, D. Favero, T.L. Kelly, Equivalences of families of stacky toric Calabi-Yau hypersurfaces. Proc. Am. Math. Soc. 146(11), 4633-4647 (2018) · Zbl 1431.14040 |
| [9] | W. Ebeling, D. Ploog, A geometric construction of Coxeter-Dynkin diagrams of bimodal singularities. Manuscripta Math. 140(1-2), 195-212 (2013) · Zbl 1264.32024 |
| [10] | D. Favero, T.L. Kelly, Fractional Calabi-Yau categories from Landau-Ginzburg models. Algebr. Geom. 5(5), 596-649 (2018) · Zbl 1423.14122 |
| [11] | W. Fulton, Introduction to Toric Varieties, 2nd edn. Ann. Math. Stu., vol. 131 (Princeton University Press, Princeton, 1997) |
| [12] | M. Kreuzer, H. Skarke, On the classification of quasihomogeneous functions. Commun. Math. Phys. 150(1), 137-147 (1992) · Zbl 0767.57019 |
| [13] | M. Kreuzer, H. Skarke, On the classification of reflexive polyhedra. Commun. Math. Phys. 185(2), 495-508 (1997) · Zbl 0894.14026 |
| [14] | M. Mase, Correspondence among families of Gorenstein \(K3\) surfaces in certain \(\mathbf{Q} \)-Fano 3-folds. Dissertation, Tokyo Metropolitan Univ., 2012. Available at https://tokyo-metro-u.repo.nii.ac.jp/?action=pages_view_main&active_action=repository_view_main_item_detail&item_id=2117&item_no=1&page_id=30&block_id=164 |
| [15] | M. Mase, A mirror duality for families of K3 surfaces associated to bimodular singularities, Manuscripta Math. 149(3-4), 389-404 (2016) · Zbl 1332.14042 |
| [16] | M. Mase, Polytope duality for families of K3 surfaces associated to transpose duality. Comment. Math. Univ. St. Pauli 65(2), 131-139 (2016) · Zbl 1378.14037 |
| [17] | M. Mase, Lattice duality for families of K3 surfaces associated to transpose duality. Manuscripta Math. https://doi.org/10.1007/s00229-017-0936-5 (online 10 April 2017) · Zbl 1387.14105 |
| [18] | M. Mase, K. Ueda, A note on bimodal singularities and mirror symmetry. Manuscripta Math. 146(1-2), 153-177 (2015) · Zbl 1405.14104 |
| [19] | M. Mase, U. Whitcher, Mirror constructions for K3 surfaces from bimodal singularities. Preprint (2022). arXiv:2206.14367 [math.AG] |
| [20] | V.V. Nikulin, Integral symmetric bilinear forms and some of their applications. Math. USSR-Izv 14, 103-167 (1980) · Zbl 0427.10014 |
| [21] | K. Nishiyama, The Jacobian fibrations on some \({K}3\) surfaces and their Mordell-Weil groups. Jpn. J. Math. 22(2), 293-347 (1996) · Zbl 0889.14015 |
| [22] | K. Oguiso, Picard numbers in a family of hyperkähler manifolds - A supplement to the article of R. Borcherds, L. Katzarkov, T. Pantev, N. I. Shepherd-Barron. Preprint (2000). arXiv.org:math/0011258 |
| [23] | F. Rohsiepe, Lattice polarized toric K3 surfaces. Preprint (2004). arXiv:hep-th/0409290 v1 |
| [24] | Sage, SageMath, the Sage Mathematics Software System (Version 8.2) (The Sage Developers, 2018). http://www.sagemath.org |
| [25] | U. Whitcher, Reflexive polytopes and lattice-polarized K3 surfaces. Calabi-Yau varieties: arithmetic, geometry and physics. Fields Inst. Monogr. 34, 65-79 (2015) · Zbl 1329.14086 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
