Laurent polynomials in mirror symmetry: why and how?

• Kasprzyk, Alexander ^[1] ; Przyjalkowski, Victor ^[2]
  1. [1] University of Nottingham

     University of Nottingham

     Reino Unido

     
  2. [2] Steklov Mathematical Institute of Russian Academy of Sciences.
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 41, Nº. Extra 2, 2022 (Ejemplar dedicado a: Special Issue on Open Questions in Geometry), págs. 481-515
• Idioma: inglés
• DOI: 10.22199/issn.0717-6279-5279
• Enlaces
  • Texto completo
• Resumen

  • We survey the approach to mirror symmetry via Laurent polynomials, outlining some of the main conjectures, problems, and questions related to the subject. We discuss: how to construct Landau–Ginzburg models for Fano varieties; how to apply them to classification problems; and how to compute invariants of Fano varieties via Landau–Ginzburg models.

• Referencias bibliográficas
  • T. Coates, A. Corti, S. Galkin, V. Golyshev, and A. M. Kasprzyk, “Mirror symmetry and Fano manifolds”, in European congress of mathematics,...
  • V. Przyjalkowski, “Toric Landau–Ginzburg models”, Uspekhi matematicheskikh nauk, vol. 73, no. 6, pp. 95–190, 2018. https://doi.org/10.4213/rm9852
  • M. Akhtar, T. Coates, S. Galkin, and A. M. Kasprzyk, “Minkowski polynomials and mutations”, Symmetry, integrability and geometry: methods...
  • T. Coates, A. M. Kasprzyk, G. Pitton, and K. Tveiten, “Maximally mutable laurent polynomials”, Proceedings of the Royal society a: mathematical,...
  • L. Katzarkov, V. Przyjalkowski, and A. Harder, “P = W Phenomena”, Matematicheskie zametki, vol. 108, no. 1, pp. 33–46, 2020. [On line]....
  • V. Lunts and V. Przyjalkowski, “Landau–Ginzburg Hodge numbers for mirrors of Del Pezzo surfaces”, Advances in mathematics, vol. 329, pp. 189–216,...
  • I. Cheltsov and V. Przyjalkowski, “Katzarkov–Kontsevich–Pantev conjecture for Fano threefolds”, 2018, arXiv: 1809.09218v1.
  • D. Auroux, L. Katzarkov, and D. Orlov, “Mirror symmetry for del pezzo surfaces: Vanishing cycles and coherent sheaves”, Inventiones mathematicae,...
  • V. S. Kulikov, “Degenerations of ?3 surfaces and Enriques surfaces”, Izvestiya akademii nauk SSSR. seriya matematicheskaya, vol. 41, no. 5,...
  • Y. I. Manin, Frobenius manifolds, quantum cohomology, and moduli spaces. Providence: AMS, 1999.
  • A. B. Givental, “Equivariant Gromov–Witten invariants”, International mathematics research notices, no. 13, pp. 613–663, 1996. https://doi.org/10.1155/S1073792896000414
  • V. V. Przyjalkowski, “Weak Landau-Ginzburg models for Smooth Fano threefolds”, Izvestiya: mathematics, vol. 77, no. 4, pp. 135–160, 2013.
  • T. Coates, A. Corti, S. Galkin, and A. Kasprzyk, “Quantum periods for 3–dimensional fano manifolds”, Geometry and topology, vol. 20, no. 1,...
  • T. Coates, A. Kasprzyk, and T. Prince, “Four-dimensional Fano Toric complete intersections”, Proceedings of the Royal society a: mathematical,...
  • T. Coates, S. Galkin, A. Kasprzyk, and A. Strangeway, “Quantum periods for certain four-dimensional Fano manifolds”, Experimental mathematics,...
  • V. Przyjalkowski and C. Shramov, “Laurent phenomenon for Landau-Ginzburg models of complete intersections in Grassmannians”, Proceedings of...
  • T. Coates, A. Kasprzyk, and T. Prince, “Laurent inversion”, Pure and applied mathematics quarterly, vol. 15, no. 4, pp. 1135–1179, 2019. https://doi.org/10.4310/pamq.2019.v15.n4.a5
  • V. V. Golyshev, “Classification problems and mirror duality” in Surveys in geometry and number theory: reports on contemporary Russian mathematics,...
  • V. Przyjalkowski, “On landau–ginzburg models for Fano varieties”, Communications in number theory and physics, vol. 1, no. 4, pp. 713–728,...
  • P. Lairez, “Computing periods of rational integrals”, Mathematics of computation, vol. 85, no. 300, pp. 1719–1752, 2015. https://doi.org/10.1090/mcom/3054
  • M. Akhtar, T. Coates, A. Corti, L. Heuberger, A. Kasprzyk, A. Oneto, A. Petracci, T. Prince, and K. Tveiten, “Mirror symmetry and the classification...
  • N. O. Ilten, J. Lewis, and V. Przyjalkowski, “Toric degenerations of Fano threefolds giving weak landau–ginzburg models”, Journal of algebra,...
  • A. M. Kasprzyk, L. Katzarkov, V. Przyjalkowski, and D. Sakovics, “Projecting Fanos in the mirror”, 2019, arXiv: 1904.02194v1.
  • A. Petracci, “An example of mirror symmetry for Fano threefolds”, in Birational geometry and moduli spaces, E. Colombo, B. Fantechi, P. Frediani,...
  • A. M. Kasprzyk and B. Nill, “Fano polytopes”, in Strings, gauge fields, and the geometry behind, A. Rebhan, L. Katzarkov, J. Knapp, R. Rashkov...
  • V. V. Przyjalkowski, “Calabi-Yau compactifications of Toric Landau-Ginzburg models for Smooth Fano threefolds”, Sbornik: mathematics, vol....
  • V. V. Przyjalkowski, “On the Calabi–yau compactifications of Toric Landau–ginzburg models for Fano complete intersections”, Mathematical zametki,...
  • V. Przyjalkowski, “On singular log Calabi–Yau compactifications of Landau–Ginzburg models”, 2021, arXiv: 2102.01388v3.
  • I. V. Dolgachev, “Mirror symmetry for lattice polarized ?3 surfaces”, 1996, arXiv: abs/alg-geom/9502005v2.
  • V. Przyjalkowski and C. Shramov, “Laurent phenomenon for Landau– Ginzburg models of complete intersections in Grassmannians of planes”, Bulletin...
  • N. O. Ilten, “Mutations of laurent polynomials and flat families with Toric fibers”, Symmetry, integrability and geometry: methods and applications,...
  • A. Petracci, “Homogeneous deformations of Toric Pairs”, Manuscripta mathematica, vol. 166, no. 1-2, pp. 37–72, 2020. https://doi.org/10.1007/s00229-020-01219-w
  • S. Galkin and A. Usnich, “Laurent phenomenon for Ginzburg–Landau potential”, IPMU preprint, 10-0100, 2010. [On line]. Available: https://bit.ly/3u5Lj2m
  • M. E. Akhtar and A. M. Kasprzyk, “Mutations of fake weighted projective planes”, Proceedings of the Edinburgh mathematical society, vol. 59,...
  • L. Katzarkov and V. Przyjalkowski, “Landau-Ginzburg models-old and new”, in Proceedings of the 18th Gökova geometry-topology conference, Gökova,...
  • P. Hacking and Y. Prokhorov, “Smoothable del pezzo surfaces with quotient singularities”, Compositio mathematica, vol. 146, no. 1, pp. 169–192,...
  • M. Akhtar and A. M. Kasprzyk, “Singularity content”, 2014, arXiv: 1401.5458v1.
  • A. M. Kasprzyk, B. Nill, and T. Prince, “Minimality and mutation-equivalence of polygons”, Forum of mathematics, Sigma, vol. 5, 2017. https://doi.org/10.1017/fms.2017.10
  • T. de Fernex and C. D. Hacon, “Rigidity properties of Fano varieties”, in Current developments in algebraic geometry, L. Caporano, J. McKernan,...
  • P. Griffiths and J. Harris, Principles of algebraic geometry. New York: Wiley-Interscience, 1978.
  • V. Przyjalkowski and C. Shramov, “On Hodge numbers of complete intersections and Landau–Ginzburg models”, International mathematics research...
  • E. Ballico, E. Gasparim, F. Rubilar, and L. A. San Martin, “KKP conjecture for minimal adjoint orbits”, 2020, arXiv: 1901.07939v3.
  • L. Katzarkov and V. Przyjalkowski, “Generalized homological mirror symmetry and cubics”, Proceedings of the Steklov institute of mathematics,...
  • L. Katzarkov, M. Kontsevich, and T. Pantev, “Bogomolov–Tian–todorov theorems for landau–ginzburg models”, Journal of differential geometry,...
  • A. Harder, “Hodge numbers of Landau–Ginzburg models”, Advances in mathematics, vol. 378, 107436, 2021. https://doi.org/10.1016/j.aim.2020.107436
  • Y. Shamoto, “Hodge–Tate conditions for Landau–Ginzburg models”, Publications of the Research institute for mathematical sciences, vol. 54,...
  • D. Auroux, “Mirror symmetry and ?-duality in the complement of an anticanonical divisor”, Journal of Gökova geometry topology, vol. 1, pp....
  • D. Auroux, “Special Lagrangian fibrations, wall-crossing, and mirror symmetry”, in Surveys in differential geometry. Geometry, analysis, and...
  • M. Abouzaid, D. Auroux, and L. Katzarkov, “Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces”, Publications...
  • M. A. de Cataldo and L. Migliorini, “The perverse filtration and the Lefschetz hyperplane theorem”, Annals of mathematics, vol. 171, no. 3,...
  • J. Kollár, “Singularities of pairs”, in Algebraic geometry-Santa Cruz 1995, vol. 2, J. Kollár, R. Lazarsfeld, D. R. Morrison, Eds. Providence:...
  • J. Kollár and S. Mori, Birational geometry of algebraic varieties. Cambridge: Cambridge University, 1998. https://doi.org/10.1017/CBO9780511662560
  • V. I. Danilov, “The geometry of toric varieties”, Uspekhi matematicheskikh nauk, vol. 33, no. 2(200), pp. 85–134, 1978. [On line]. Available:...
  • I. Cheltsov and V. Przyjalkowski, “Fibers over infinity of Landau–Ginzburg models”, 2020, arXiv: 2005.01534v1.
  • A. Höring and C. Voisin, “Anticanonical divisors and curve classes on Fano manifolds”, Pure and applied mathematics quarterly, vol. 7, no....
  • A. Höring and R. Śmiech, “Anticanonical System of Fano fivefolds”, Mathematische nachrichten, vol. 293, no. 1, pp. 115–119, 2019. https://doi.org/10.1002/mana.201900311
  • Y. Kawamata, “On effective non-vanishing and base-point-freeness”, Asian journal of mathematics, vol. 4, no. 1, pp. 173–182, 2000. https://doi.org/10.4310/ajm.2000.v4.n1.a11
  • J. Kollár and C. Xu, “The dual complex of Calabi–Yau Pairs”, Inventiones mathematicae, vol. 205, no. 3, pp. 527–557, 2016. https://doi.org/10.1007/s00222-015-0640-6