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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.
DOI:
https://doi.org/10.4213/im8018
 Full text:
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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
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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
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V. Gorbounov, V. Petrov, “Schubert calculus and singularity theory”, J. Geom. Phys., 62:2 (2012), 352–360
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N. O. Ilten, J. Lewis, V. Przyjalkowski, “Degenerations of Fano threefolds giving weak Landau-Ginzburg models”, J. Algebra, 374 (2013), 104–121
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J. A. Christophersen, N. O. Ilten, “Degenerations to unobstructed Fano Stanley–Reisner schemes”, Math. Z., 278:1-2 (2014), 131–148
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A. Iliev, L. Katzarkov, V. Przyjalkowski, “Double solids, categories and non-rationality”, Proc. Edinb. Math. Soc. (2), 57:1 (2014), 145–173
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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
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V. Gorbounov, M. Smirnov, “Some remarks on Landau-Ginzburg potentials for odd-dimensional quadrics”, Glasg. Math. J., 57:3 (2015), 481–507
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G. Kapustka, “Projections of del Pezzo surfaces and Calabi-Yau threefolds”, Adv. Geom., 15:2 (2015), 143–158
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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
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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
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V. V. Przyjalkowski, “Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds”, Sb. Math., 208:7 (2017), 992–1013
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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
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V. V. Przyjalkowski, “On the Calabi–Yau Compactifications of Toric Landau–Ginzburg Models for Fano Complete Intersections”, Math. Notes, 103:1 (2018), 104–110
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V. Lunts, V. Przyjalkowski, “Landau-Ginzburg Hodge numbers for mirrors of del Pezzo surfaces”, Adv. Math., 329 (2018), 189–216
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V. V. Przyjalkowski, “Toric Landau–Ginzburg models”, Russian Math. Surveys, 73:6 (2018), 1033–1118
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Przyjalkowski V., Shramov C., “Nef Partitions For Codimension 2 Weighted Complete Intersections”, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 19:3 (2019), 827–845
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L. Katzarkov, V. V. Przyjalkowski, A. Harder, “$\mathrm P=\mathrm W$ Phenomena”, Math. Notes, 108:1 (2020), 39–49
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