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Izv. RAN. Ser. Mat., 2013, Volume 77, Issue 4, Pages 135–160 (Mi izv8018)  

This article is cited in 17 scientific papers (total in 17 papers)

Weak Landau–Ginzburg models for smooth Fano threefolds

V. V. Przyjalkowski

Steklov Mathematical Institute of the Russian Academy of Sciences

Abstract: We consider Landau–Ginzburg models for smooth Fano threefolds of the principal series and prove that they can be represented by Laurent polynomials. We check that these models can be compactified to open Calabi–Yau varieties. In the spirit of Katzarkov's programme we prove that the numbers of irreducible components of the central fibres of compactifications of these pencils are equal to the dimensions of intermediate Jacobians of the corresponding Fano varieties plus 1. In particular, these numbers are independent of the choice of compactification. We state most of the known methods for finding Landau–Ginzburg models in terms of Laurent polynomials. We discuss the Laurent polynomial representation of the Landau–Ginzburg models of Fano varieties and state some related problems.

Keywords: weak Landau–Ginzburg models, Fano varieties, toric degeneration, intermediate Jacobian.

Funding Agency Grant Number
Austrian Science Fund P20778
Russian Foundation for Basic Research 11-01-00336-a
Ministry of Education and Science of the Russian Federation МК-1192.2012.1
Dynasty Foundation
National Science Foundation DMS-0854977


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English version:
Izvestiya: Mathematics, 2013, 77:4, 772–794

Bibliographic databases:

ArXiv: 0902.4668
UDC: 512.776
MSC: 14J33, 14J45, 14N35
Received: 26.06.2012
Revised: 15.10.2012

Citation: V. V. Przyjalkowski, “Weak Landau–Ginzburg models for smooth Fano threefolds”, Izv. RAN. Ser. Mat., 77:4 (2013), 135–160; Izv. Math., 77:4 (2013), 772–794

Citation in format AMSBIB
\by V.~V.~Przyjalkowski
\paper Weak Landau--Ginzburg models for smooth Fano threefolds
\jour Izv. RAN. Ser. Mat.
\yr 2013
\vol 77
\issue 4
\pages 135--160
\jour Izv. Math.
\yr 2013
\vol 77
\issue 4
\pages 772--794

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    This publication is cited in the following articles:
    1. V. Przyjalkowski, “Hori-Vafa mirror models for complete intersections in weighted projective spaces and weak Landau-Ginzburg models”, Cent. Eur. J. Math., 9:5 (2011), 972–977  crossref  mathscinet  zmath  isi  elib  scopus
    2. V. Gorbounov, V. Petrov, “Schubert calculus and singularity theory”, J. Geom. Phys., 62:2 (2012), 352–360  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. N. O. Ilten, J. Lewis, V. Przyjalkowski, “Degenerations of Fano threefolds giving weak Landau-Ginzburg models”, J. Algebra, 374 (2013), 104–121  crossref  mathscinet  zmath  isi  elib  scopus
    4. J. A. Christophersen, N. O. Ilten, “Degenerations to unobstructed Fano Stanley–Reisner schemes”, Math. Z., 278:1-2 (2014), 131–148  crossref  mathscinet  zmath  isi  elib  scopus
    5. A. Iliev, L. Katzarkov, V. Przyjalkowski, “Double solids, categories and non-rationality”, Proc. Edinb. Math. Soc. (2), 57:1 (2014), 145–173  crossref  mathscinet  zmath  isi  scopus
    6. V. V. Przyjalkowski, C. A. Shramov, “Laurent phenomenon for Landau–Ginzburg models of complete intersections in Grassmannians”, Proc. Steklov Inst. Math., 290:1 (2015), 91–102  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    7. V. Gorbounov, M. Smirnov, “Some remarks on Landau-Ginzburg potentials for odd-dimensional quadrics”, Glasg. Math. J., 57:3 (2015), 481–507  crossref  mathscinet  zmath  isi  scopus
    8. G. Kapustka, “Projections of del Pezzo surfaces and Calabi-Yau threefolds”, Adv. Geom., 15:2 (2015), 143–158  crossref  mathscinet  zmath  isi  scopus
    9. T. Coates, A. Kasprzyk, T. Prince, “Four-dimensional Fano toric complete intersections”, Proc. of The Royal Society A. Math., Phys. and Eng. Sci., 471 (2015), 2175  crossref  mathscinet  zmath  elib  scopus
    10. V. V. Golyshev, D. Zagier, “Proof of the gamma conjecture for Fano 3-folds of Picard rank 1”, Izv. Math., 80:1 (2016), 24–49  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. V. V. Przyjalkowski, “Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds”, Sb. Math., 208:7 (2017), 992–1013  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    12. C. Shramov, V. Przyjalkowski, “Laurent phenomenon for Landau-Ginzburg models of complete intersections in Grassmannians of planes”, Bull. Korean Math. Soc., 54:5 (2017), 1527–1575 , arXiv: 1409.3729  mathnet  crossref  mathscinet  isi  scopus
    13. V. V. Przyjalkowski, “On the Calabi–Yau Compactifications of Toric Landau–Ginzburg Models for Fano Complete Intersections”, Math. Notes, 103:1 (2018), 104–110  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    14. V. Lunts, V. Przyjalkowski, “Landau-Ginzburg Hodge numbers for mirrors of del Pezzo surfaces”, Adv. Math., 329 (2018), 189–216  crossref  mathscinet  zmath  isi  scopus
    15. V. V. Przyjalkowski, “Toric Landau–Ginzburg models”, Russian Math. Surveys, 73:6 (2018), 1033–1118  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    16. Przyjalkowski V., Shramov C., “Nef Partitions For Codimension 2 Weighted Complete Intersections”, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 19:3 (2019), 827–845  isi
    17. L. Katzarkov, V. V. Przyjalkowski, A. Harder, “$\mathrm P=\mathrm W$ Phenomena”, Math. Notes, 108:1 (2020), 39–49  mathnet  crossref  crossref  mathscinet  isi
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