Hostname: page-component-857557d7f7-ktsnh Total loading time: 0 Render date: 2025-11-30T11:25:34.616Z Has data issue: false hasContentIssue false
  • English
  • Français

Toric Degenerations and Laurent Polynomials Related to Givental's Landau–Ginzburg Models

Published online by Cambridge University Press:  20 November 2018

Charles F. Doran  and
Andrew Harder
Charles F. Doran
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1 e-mail: charles.doran@ualberta.ca aharder@ualberta.ca
Andrew Harder
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1 e-mail: charles.doran@ualberta.ca aharder@ualberta.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For an appropriate class of Fano complete intersections in toric varieties, we prove that there is a concrete relationship between degenerations to specific toric subvarieties and expressions for Givental's Landau–Ginzburg models as Laurent polynomials. As a result, we show that Fano varieties presented as complete intersections in partial flag manifolds admit degenerations to Gorenstein toric weak Fano varieties, and their Givental Landau–Ginzburg models can be expressed as corresponding Laurent polynomials.

We also use this to show that all of the Laurent polynomials obtained by Coates, Kasprzyk and Prince by the so–called Przyjalkowski method correspond to toric degenerations of the corresponding Fano variety. We discuss applications to geometric transitions of Calabi–Yau varieties.

Keywords

14M2514J3214J3314J45Fano varietiesLandau-Ginzburg modelsCalabi-Yau varietiestoric varieties

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 68 , Issue 4 , 01 August 2016 , pp. 784 - 815
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Akhtar, M., Coates, T., Galkin, S., and Kasprzyk, A., Minkowski polynomials and mutations. SIGMA Symmetry Integrability Geom. Methods Appl. 8(2012),094. http://dx.doi.Org/10.3842/SICMA.2012.094 Google Scholar
[2] Aspinwall, P. S., Greene, B. R., and Morrison, D. R., The monomial-divisor mirror map. Internat. Math. Res. Notices 1993, no. 12, 319–337.http://dx.doi.Org/10.1155/S1073792893000376 Google Scholar
[3] Batyrev, V. and Nill, B., Combinatorial aspects of mirror symmetry. In: Integer points in polyhedra—geometry, number theory, representation theory, algebra, optimization, statistics, Contemp. Math., 452, American Mathematical Society, Providence, RI, 2008, pp. 35–66.http://dx.doi.Org/10.1090/conm/452/08770 Google Scholar
[4] Batyrev, V. V., Toric degenerations ofFano varieties and constructing mirror manifolds. In: The Fano Conference, Univ. Torino, Turin, 2004, pp. 109–122.Google Scholar
[5] Batyrev, V. V. and Borisov, L. A.,On Calabi-Yau complete intersections in toric varieties. In: Higher-dimensional complex varieties (Trento, 1994), de Gruyter, Berlin, 1996, pp. 39–65.Google Scholar
[6] Batyrev, V. V., Ciocan-Fontanine, I., Kim, B., and van Straten, D..Mirror symmetry and toric degenerations of partial flag manifolds. Acta Math. 184(2000), no. 1,1–39. http://dx.doi.org/10.1007/BF02392780 Google Scholar
[7] Brion, M., Lectures on the geometry of flag varieties. In: Topics in cohomological studies of algebraic varieties, Trends Math., Birkhâuser, Basel, 2005, pp. 33–85.http://dx.doi.Org/10.1007/3-7643-7342-3_2 Google Scholar
[8] Coates, T., Kasprzyk, A., and Prince, T., Four-dimensional Fano toric complete intersections.Proc. A. 471(2015), no. 2175,20140704.http://dx.doi.Org/10.1098/rspa.2O14.0704 Google Scholar
[9] Cox, D. A., Little, J. B., and Schenck, H. K., Toric varieties.Graduate Studies in Mathematics, 124, American Mathematical Society, Providence, RI,2011.http://dx.doi.Org/10.1090/gsm/124 Google Scholar
[10] Eguchi, T., Hori, K., and Xiong, C.-S., Gravitational quantum cohomology. Internat. J. Modern Phys. A 12(1997), no. 9, 1743–1782.http://dx.doi.Org/10.1142/S0217751X97001146 Google Scholar
[11 Fischer, K. G. and Shapiro, J., Mixed matrices and binomial ideals. J. Pure Appl. Algebra 113(1996), no. 1, 39–54.http://dx.doi.Org/10.1016/0022-4049(95)00144-1 Google Scholar
[12] Fredrickson, K., Mirror transitions and the Batyrev-Borisov construction. Beitr. Algebra Geom. 56(2015), no. 2, 677–693.http://dx.doi.org/10.1007/s13366-015-0242-x Google Scholar
[13] Galkin, S. and Usnich, A., Mutations of potentials. IPMU preprint, http://www.mccme.ru/∼galkin/papers/ipmu-10-0100.pdf Google Scholar
[14] Givental, A., A mirror theorem for toric complete intersections. In: Topological field theory, primitive forms and related topics (Kyoto, 1996), Progr. Math., 160, Birkhâuser Boston, Boston, MA, 1998, pp. 141–175.Google Scholar
[15] Gonciulea, N. and Lakshmibai, V., Degenerations of flag and Schubert varieties to toric varieties. Transform. Groups 1(1996), no. 3, 215–248.http://dx.doi.Org/10.1007/BF02549207 Google Scholar
[16] Hori, K. and Vafa, C., Mirror symmetry. arxiv:hep-th/OOO2222v3Google Scholar
[17] Owen Ilten, N., Lewis, J., and Przyjalkowski, V., Toric degenerations ofFano threefolds giving weak Landau-Ginzburg models. J. Algebra 374(2013), 104–121. http://dx.doi.Org/10.1016/j.jalgebra.2O12.11.002 Google Scholar
[18] Katzarkov, L., Kontsevich, M., and Pantev, T., Hodge theoretic aspects of mirror symmetry. In: From Hodge theory to integrability and TQFT tt*-geometry, Proc. Sympos. Pure Math., 78, American Mathematical Society, Providence, RI, 2008, pp. 87–174. Google Scholar
[19] Katzarkov, L., Kontsevich, M., and Pantev, T., Bogomolov-Tian-Todorov theorems for Landau-Ginzburgmodels. arxiv:1409.5996v1Google Scholar
[20] Katzarkov, L. and Przyjalkowski, V., Landau-Ginzburg models—old and new. In: Proceedings of the Gôkova Geometry-Topology Conference 2011, Int. Press, Somerville, MA, 2012, pp. 97–124.Google Scholar
[21] Mavlyutov, A., Deformations oftoric varieties via Minkowski sum decompositions of polyhedralcomplexes. arxiv:0902.0967v3Google Scholar
[22] Przyjalkowski, V., Hori-Vafa mirror models for complete intersections in weighted projective spaces and weak Landau-Ginzburg models. Cent. Eur. J. Math. 9(2011), no. 5, 972–977.http://dx.doi.org/10.2478/s11533-011-0070-7 Google Scholar
[23] Przyjalkowski, V. and Shramov, C., Laurent phenomenon for Landau-Ginzburg models of complete intersections in Grassmannians of planes. arxiv:1409.3729v1Google Scholar