Equivalences of families of stacky toric Calabi-Yau hypersurfaces
HTML articles powered by AMS MathViewer
- by Charles F. Doran, David Favero and Tyler L. Kelly
- Proc. Amer. Math. Soc. 146 (2018), 4633-4647
- DOI: https://doi.org/10.1090/proc/14154
- Published electronically: August 10, 2018
- PDF | Request permission
Abstract:
Given the same anti-canonical linear system on two distinct toric varieties, we provide a derived equivalence between partial crepant resolutions of the corresponding stacky hypersurfaces. The applications include: a derived unification of toric mirror constructions, calculations of Picard lattices for linear systems of quartics in $\mathbf {P}^3$, and a birational reduction of Reid’s list to 81 families.References
- Michela Artebani, Paola Comparin, and Robin Guilbot, Families of Calabi-Yau hypersurfaces in $\Bbb {Q}$-Fano toric varieties, J. Math. Pures Appl. (9) 106 (2016), no. 2, 319–341 (English, with English and French summaries). MR 3515305, DOI 10.1016/j.matpur.2016.02.012
- Paul S. Aspinwall and M. Ronen Plesser, General mirror pairs for gauged linear sigma models, J. High Energy Phys. 11 (2015), 029, front matter+32. MR 3455065, DOI 10.1007/JHEP11(2015)029
- M. Ballard, D. Favero, and L. Katzarkov, Variation of Geometric Invariant Theory quotients and derived categories, arXiv:1203.6643, to appear in Crelle.
- Victor V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994), no. 3, 493–535. MR 1269718
- Victor V. Batyrev and Lev A. Borisov, Mirror duality and string-theoretic Hodge numbers, Invent. Math. 126 (1996), no. 1, 183–203. MR 1408560, DOI 10.1007/s002220050093
- Sarah-Marie Belcastro, Picard lattices of families of $K3$ surfaces, Comm. Algebra 30 (2002), no. 1, 61–82. MR 1880661, DOI 10.1081/AGB-120006479
- Per Berglund and Tristan Hübsch, A generalized construction of mirror manifolds, Nuclear Phys. B 393 (1993), no. 1-2, 377–391. MR 1214325, DOI 10.1016/0550-3213(93)90250-S
- Lev A. Borisov, Berglund-Hübsch mirror symmetry via vertex algebras, Comm. Math. Phys. 320 (2013), no. 1, 73–99. MR 3046990, DOI 10.1007/s00220-013-1717-y
- Ugo Bruzzo and Antonella Grassi, Picard group of hypersurfaces in toric 3-folds, Internat. J. Math. 23 (2012), no. 2, 1250028, 14. MR 2890472, DOI 10.1142/S0129167X12500280
- Patrick Clarke, A proof of the birationality of certain BHK-mirrors, Complex Manifolds 1 (2014), no. 1, 45–51. MR 3333467, DOI 10.2478/coma-2014-0003
- Patrick Clarke, Birationality and Landau-Ginzburg Models, Comm. Math. Phys. 353 (2017), no. 3, 1241–1260. MR 3652490, DOI 10.1007/s00220-017-2830-0
- Patrick Clarke, Duality for toric Landau-Ginzburg models, Adv. Theor. Math. Phys. 21 (2017), no. 1, 243–287. MR 3636695, DOI 10.4310/ATMP.2017.v21.n1.a5
- Adrian Clingher and Charles F. Doran, Lattice polarized K3 surfaces and Siegel modular forms, Adv. Math. 231 (2012), no. 1, 172–212. MR 2935386, DOI 10.1016/j.aim.2012.05.001
- David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322, DOI 10.1090/gsm/124
- Charles Doran, Brian Greene, and Simon Judes, Families of quintic Calabi-Yau 3-folds with discrete symmetries, Comm. Math. Phys. 280 (2008), no. 3, 675–725. MR 2399610, DOI 10.1007/s00220-008-0473-x
- D. Favero and T. L. Kelly, Derived Categories of BHK Mirrors, arXiv:1602.05876.
- D. Favero and T. L. Kelly, Fractional Calabi-Yau Categories via Landau-Ginzburg Models, arXiv:1610:04918, to appear in Algebraic Geometry.
- David Favero and Tyler L. Kelly, Proof of a conjecture of Batyrev and Nill, Amer. J. Math. 139 (2017), no. 6, 1493–1520. MR 3730928, DOI 10.1353/ajm.2017.0038
- A. R. Iano-Fletcher, Working with weighted complete intersections, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 101–173. MR 1798982
- D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/
- B. R. Greene and M. R. Plesser, Duality in Calabi-Yau moduli space, Nuclear Phys. B 338 (1990), no. 1, 15–37. MR 1059831, DOI 10.1016/0550-3213(90)90622-K
- Daniel Halpern-Leistner, The derived category of a GIT quotient, J. Amer. Math. Soc. 28 (2015), no. 3, 871–912. MR 3327537, DOI 10.1090/S0894-0347-2014-00815-8
- Manfred Herbst and Johannes Walcher, On the unipotence of autoequivalences of toric complete intersection Calabi-Yau categories, Math. Ann. 353 (2012), no. 3, 783–802. MR 2923950, DOI 10.1007/s00208-011-0704-x
- Yujiro Kawamata, Log crepant birational maps and derived categories, J. Math. Sci. Univ. Tokyo 12 (2005), no. 2, 211–231. MR 2150737
- Tyler L. Kelly, Berglund-Hübsch-Krawitz mirrors via Shioda maps, Adv. Theor. Math. Phys. 17 (2013), no. 6, 1425–1449. MR 3262528
- Masanori Kobayashi and Makiko Mase, Isomorphism among families of weighted $K3$ hypersurfaces, Tokyo J. Math. 35 (2012), no. 2, 461–467. MR 3058718, DOI 10.3836/tjm/1358951330
- Marc Krawitz, FJRW rings and Landau-Ginzburg mirror symmetry, ProQuest LLC, Ann Arbor, MI, 2010. Thesis (Ph.D.)–University of Michigan. MR 2801653
- Maximilian Kreuzer and Harald Skarke, On the classification of quasihomogeneous functions, Comm. Math. Phys. 150 (1992), no. 1, 137–147. MR 1188500
- Zhan Li, On the birationality of complete intersections associated to nef-partitions, Adv. Math. 299 (2016), 71–107. MR 3519464, DOI 10.1016/j.aim.2016.05.006
- Joshua P. Mullet, Toric Calabi-Yau hypersurfaces fibered by weighted $K3$ hypersurfaces, Comm. Anal. Geom. 17 (2009), no. 1, 107–138. MR 2495835, DOI 10.4310/CAG.2009.v17.n1.a5
- Miles Reid, The moduli space of $3$-folds with $K=0$ may nevertheless be irreducible, Math. Ann. 278 (1987), no. 1-4, 329–334. MR 909231, DOI 10.1007/BF01458074
- F. Rohsiepe, Lattice polarized toric K3 surfaces, arXiv:hep-th/0409290.
- SageMath, the Sage Mathematics Software System (Version 8.1), The Sage Developers, 2018, http://www.sagemath.org.
- Mark Shoemaker, Birationality of Berglund-Hübsch-Krawitz mirrors, Comm. Math. Phys. 331 (2014), no. 2, 417–429. MR 3238520, DOI 10.1007/s00220-014-2121-y
- E. B. Vinberg, On the algebra of Siegel modular forms of genus 2, Trans. Moscow Math. Soc. , posted on (2013), 1–13. MR 3235787, DOI 10.1090/s0077-1554-2014-00217-x
- Jarosław Włodarczyk, Decomposition of birational toric maps in blow-ups & blow-downs, Trans. Amer. Math. Soc. 349 (1997), no. 1, 373–411. MR 1370654, DOI 10.1090/S0002-9947-97-01701-7
Bibliographic Information
- Charles F. Doran
- Affiliation: Department of Mathematics, University of Alberta, Edmonton, Alberta Canada
- MR Author ID: 643024
- Email: doran@math.ualberta.ca
- David Favero
- Affiliation: Department of Mathematics, University of Alberta, Edmonton, Alberta Canada – and – Korea Institute for Advanced Study, Seoul, Republic of Korea
- MR Author ID: 739092
- ORCID: 0000-0002-6376-6789
- Email: favero@ualberta.ca
- Tyler L. Kelly
- Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB United Kingdom
- MR Author ID: 874289
- Email: tlk20@dpmms.cam.ac.uk
- Received by editor(s): October 3, 2017
- Received by editor(s) in revised form: February 20, 2018, and March 2, 2018
- Published electronically: August 10, 2018
- Additional Notes: The first author was supported by NSERC, PIMS, and a McCalla professorship at the University of Alberta.
The second author was supported by NSERC through a Discovery Grant and as a Canada Research Chair.
The third author was supported in part by NSF Grant # DMS-1401446 and EPSRC Grant EP/N004922/1. - Communicated by: Lev Borisov
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 4633-4647
- MSC (2010): Primary 14M25; Secondary 14C22, 14J33, 14J32, 14J28
- DOI: https://doi.org/10.1090/proc/14154
- MathSciNet review: 3856133