Birationality and Landau-Ginzburg models. (English) Zbl 1428.14074

Summary: We introduce a new technique for approaching birationality questions that arise in the mirror symmetry of complete intersections in toric varieties. As an application we answer affirmatively and conclusively the question of V. Batyrev and B. Nill [Contemp. Math. 452, 35–66 (2008; Zbl 1161.14037)]. about the birationality of Calabi-Yau families associated to multiple mirror nef-partitions. This completes the progress in this direction made by Z. Li’s breakthrough [Adv. Math. 299, 71–107 (2016; Zbl 1360.14040)]. In the process, we obtain results in the theory of L. Borisov’s nef-partitions [“Towards the mirror symmetry for Calabi-Yau complete intersections in Gorenstein toric Fano varieties”, Preprint, arXiv:alg-geom/9310001] and provide new insight into the geometric content of the multiple mirror phenomenon.


14J33 Mirror symmetry (algebro-geometric aspects)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14E07 Birational automorphisms, Cremona group and generalizations
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
81T13 Yang-Mills and other gauge theories in quantum field theory
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
Full Text: DOI arXiv


[1] Batyrev V.V.: Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Algebr. Geom. 3(3), 493-535 (1994) · Zbl 0829.14023
[2] Batyrev, V.V.: Birational Calabi-Yau n-folds have equal Betti numbers. In: New Trends in Algebraic Geometry (Warwick, 1996), volume 264 of London Mathematical Society Lecture Note Series, pp. 1-11. Cambridge University Press, Cambridge (1999) · Zbl 0955.14028
[3] Batyrev, V.V., Borisov, L.A.: On Calabi-Yau complete intersections in toric varieties. In: Higher-Dimensional Complex Varieties (Trento, 1994), pp. 39-65. de Gruyter, Berlin (1996) · Zbl 0908.14015
[4] Batyrev, V.V., Borisov, L.A.: Dual cones and mirror symmetry for generalized Calabi-Yau manifolds. In: Mirror Symmetry, II, volume 1 of AMS/IP Studies in Advanced Mathematics, pp. 71-86. American Mathematical Society, Providence (1997) · Zbl 0927.14019
[5] Ballard, M., Favero, D., Katzarkov, L.: Variation of geometric invariant theory quotients and derived categories. J. Reine Angew. Math. (Crelles J.) (2016). doi:10.1515/crelle-2015-0096 · Zbl 1432.14015
[6] Berglund P., Hübsch T.: A generalized construction of mirror manifolds. Nucl. Phys. B. 393(1-2), 377-391 (1993) · Zbl 1245.14039
[7] Batyrev, V., Benjamin, N.: Combinatorial aspects of mirror symmetry. In: Barvinok, A., Beck, M., Haase, C., Reznick, B., Welker, V. (eds.) Integer Points in Polyhedra—Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, volume 452 of Contemporary Mathematics, pp. 35-66. American Mathematical Society, Providence (2008) · Zbl 1161.14037
[8] Bondal A., Orlov D.: Reconstruction of a variety from the derived category and groups of autoequivalences. Compos. Math. 125(3), 327-344 (2001) · Zbl 0994.18007
[9] Borisov, L.: Towards the mirror symmetry for Calabi-Yau complete intersections in Gorenstein toric Fano varieties (1993). arXiv:alg-geom/9310001 · Zbl 1375.14133
[10] Clarke, P.: Duality for toric Landau-Ginzburg models (2008). arXiv:0803.0447 · Zbl 1386.81130
[11] Clarke P.: A proof of the birationality of certain BHK-mirrors. Complex Manifolds 1, 45-51 (2014) · Zbl 1320.32032
[12] Clarke, P.: Birationality and Landau-Ginzburg models (2016). arXiv:1608.07917. · Zbl 1428.14074
[13] Clarke P.: Dual fans and mirror symmetry. Adv. Math. 301, 902-933 (2016) · Zbl 1375.14133
[14] Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties, volume 124 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2011) · Zbl 1223.14001
[15] Doran, C.F., Favero, D., Kelly, T.L.: Equivalences of families of stacky toric Calabi-Yau hypersurfaces (2015). arXiv:1503.04888 · Zbl 1431.14040
[16] Dixon, L.J.: Some world sheet properties of superstring compactifications, on orbifolds and otherwise. In: Proceedings, Summer Workshop in High-energy Physics and Cosmology: Superstrings, Unified Theories and Cosmology: Trieste, Italy, 29 June-7 Aug 1987 · Zbl 1271.14076
[17] Favero, D., Kelly, T.L.: Proof of a conjecture of Batyrev and Nill. Am. J. Math. (2014). arXiv:1412.1354 · Zbl 1390.14124
[18] Favero, D., Kelly, T.L.: Derived categories of BHK mirrors (2016). arXiv:1602.05876 · Zbl 1444.14076
[19] Frobenius, G.: Üeber Matrizen aus nicht negativen Elementen. Sitz. ber. K. Preuss. Akad. Wiss. 23, 456-477 (1912) · JFM 43.0204.09
[20] Fulton, W.: Introduction to Toric Varieties, volume 131 of Annals of Mathematics Studies. Princeton University Press, Princeton. The William H. Roever Lectures in Geometry (1993) · Zbl 0813.14039
[21] Givental, A.: A mirror theorem for toric complete intersections. In: Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996), volume 160 of Progress in Mathematics, pp. 141-175. Birkhäuser, Boston (1998) · Zbl 0936.14031
[22] Greene B.R., Plesser M.R.: Duality in Calabi-Yau moduli space. Nucl. Phys. B. 338(1), 15-37 (1990)
[23] Herbst, M., Hori, K., Page, D.: B-type D-branes in toric Calabi-Yau varieties. In Homological Mirror Symmetry, volume 757 of Lecture Notes in Physics, pp. 27-44. Springer, Berlin (2009) · Zbl 1162.81033
[24] Hori, K., Vafa, C.: Mirror symmetry (2000). arXiv:hep-th/0002222 · Zbl 1044.14018
[25] Kelly Tyler L.: Berglund-Hübsch-Krawitz mirrors via Shioda maps. Adv. Theor. Math. Phys. 17(6), 1425-1449 (2013) · Zbl 1316.14076
[26] Kontsevich, M.: Homological algebra of mirror symmetry. In: Proceedings of the International Congress of Mathematicians, vols. 1, 2(Zürich, 1994), pp. 120-139. Birkhäuser, Basel (1995) · Zbl 0846.53021
[27] Krawitz, M.: FJRW rings and Landau-Ginzburg mirror symmetry. ProQuest LLC, Ann Arbor. Thesis (Ph.D.), University of Michigan (2010). arXiv:0906.0796 · Zbl 1250.81087
[28] Kreuzer M., Riegler E., Sahakyan David A.: Toric complete intersections and weighted projective space. J. Geom. Phys. 46(2), 159-173 (2003) · Zbl 1061.14037
[29] Li Z.: On the birationality of complete intersections associated to nef-partitions. Adv. Math. 299, 71-107 (2016) · Zbl 1360.14040
[30] Lerche W., Vafa C., Warner N.P.: Chiral rings in N = 2 superconformal theories. Nucl. Phys. B 324, 427-474 (1989)
[31] Meyer, C.: Matrix analysis and applied linear algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia. With 1 CD-ROM (Windows, Macintosh and UNIX) and a solutions manual (iv+171) (2000)
[32] Nill B., Schepers J.: Gorenstein polytopes and their stringy E-functions. Math. Ann. 355(2), 457-480 (2013) · Zbl 1271.14076
[33] Oda, T.: Torus Embeddings and Applications, volume 57 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Tata Institute of Fundamental Research, Bombay; by Springer, Berlin. Based on joint work with Katsuya Miyake (1978) · Zbl 0417.14043
[34] Perron O.: Zur Theorie der Matrices. Math. Ann. 64(2), 248-263 (1907) · JFM 38.0202.01
[35] Shoemaker M.: Birationality of Berglund—Hübsch-Krawitz mirrors. Commun. Math. Phys. 331(2), 417-429 (2014) · Zbl 1395.14034
[36] Witten, E.: Phases of N=2 theories in two dimensions. In: Mirror Symmetry, II, volume 1 of AMS/IP Studies in Advanced Mathematics, pp. 143-211. American Mathematical Society, Providence (1997) · Zbl 0910.14019
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.