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Weak Landau-Ginzburg models for smooth Fano threefolds. (English. Russian original) Zbl 1281.14033
Izv. Math. 77, No. 4, 772-794 (2013); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 77, No. 4, 135-160 (2013).
Let \(X\) be a smooth Fano variety of dimension \(n\) and Picard rank 1, and \(QH^*(X)\) be the quantum cohomology of \(X\). For \(H=-K_X\) define \(QH_H^*(X)\subset QH^*(X)\) the minimal subring containing \(H\). Then \(X\) is called quantum minimal if the dimension of \(QH_H^*\) over the Novikov ring is equal to \(n+1\). The fundamental term of the regularized \(I\)-series of \(X\) denoted by \(\tilde{I}_{H^0}^X\in \mathbb{C}[[t]]\) is defined in terms of the generating function of genus 0 Gromov-Witten invariants of the quantum minimal variety \(X\), having the descendants of \(H^0\) (dual of the fundamental class of \(X\)) as insertions. For the regular function \(f\in \mathbb{C}[x_1^{\pm1},\dots,x_n^{\pm1}]\) defined on the torus \(\mathbb{G}^n_m\), we denote by \(\Phi_f \in \mathbb{C}[[t]]\) the constant-term series of \(f\) (with coefficient of \(t^i\) equal to constant term of \(f^i\)). Then \(f\) is called a weak Landau-Ginzburg model for \(X\) if \(\Phi_f=\tilde{I}_{H^0}^X\), and furthermore if there is a fiberwise smooth (open) Calabi-Yau compactification of the family \(f:\mathbb{C}^{*n}\to \mathbb{C}\) model for \(X\) if \(\Phi_f=\tilde{I}_{H^0}^X\). Finding a weak Landau-Ginzburg model for \(X\) provides a coincidence of the invariants of categories involved in the Homological Mirror Symmetry.
There are 17 families of smooth Fano threefolds of Picard rank 1. The paper under review provides a weak Landau-Ginzburg model as a Laurent polynomial in three variables as discussed above for all 17 families, and surveys all the known methods for finding such Laurent polynomials. These Laurent polynomials are shown to have Calabi-Yau compactifications to families of \(K3\)-surfaces. The Calabi-Yau compactifications of any of these families differ by flops. Following the ideas of Katzarkov, the paper under review proves that the numbers of irreducible components of central fibers of these Calabi-Yau compactifications are always 1 less than the Hodge number \(h^{12}(X)\), and in particular they are independent of the choice of the compactification. This result enables one to reconstruct the Hodge numbers of these Fano threefolds.

MSC:
14J33 Mirror symmetry (algebro-geometric aspects)
14J45 Fano varieties
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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References:
[1] M. Kontsevich, “Homological algebra of mirror symmetry”, Proceedings of the international congress of mathematicians (Zu\"rich, 1994), Birkhaüzer, Basel, 1995, 120 – 139 · Zbl 0846.53021
[2] L. Katzarkov, M. Kontsevich, T. Pantev, “Hodge theoretic aspects of mirror symmetry”, From Hodge theory to integrability and TQFT \(tt^*\)-geometry (Augsburg, Germany, 2007), Proc. Sympos. Pure Math., 78, Amer. Math. Soc., Providence, RI, 2008, 87 – 174 · Zbl 1206.14009
[3] V. Przyjalkowski, “On Landau – Ginzburg models for Fano varieties”, Commun. Number Theory Phys., 1:4 (2008), 713 – 728 · Zbl 1194.14065
[4] В. А. Исковских, “Трехмерные многообразия Фано. I”, Изв. АН СССР. Сер. матем., 41:3 (1977), 516 – 562 · Zbl 0363.14010
[5] “II”, Изв. АН СССР. Сер. матем., 42:3 (1978), 506 – 549 · Zbl 0407.14016
[6] V. A. Iskovskih, “Fano 3-folds. I”, Math. USSR-Izv., 11:3 (1977), 485 – 527 · Zbl 0382.14013
[7] “II”, Math. USSR-Izv., 12:3 (1978), 469 – 506 · Zbl 0424.14012
[8] C. van Enckevort, D. van Straten, “Monodromy calculations of fourth order equations of Calabi – Yau type”, Mirror symmetry V, AMS/IP Stud. Adv. Math., 38, Amer. Math. Soc., Providence, RI, 2006, 539 – 559 · Zbl 1117.14043
[9] N. O. Ilten, J. Lewis, V. Przyjalkowski, “Toric degenerations of Fano threefolds giving weak Landau – Ginzburg models”, J. Algebra, 374 (2013), 104 – 121 · Zbl 1270.14020
[10] L. Katzarkov, V. Przyjalkowski, “Landau – Ginzburg models – old and new”, Proceedings of the 18th Gokova geometry-topology conference, International Press, Somerville, MA, 2012, 97 – 124 · Zbl 1360.81003
[11] C. Doran, A. Harder, L. Katzarkov, J. Lewis, V. Przyjalkowski, “Modularity of Fano threefolds” (to appear)
[12] M. Abouzaid, D. Auroux, L. Katzarkov, Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces, arXiv: 1205.0053 · Zbl 1368.14056
[13] I. Cheltsov, L. Katzarkov, V. Przyjalkowski, “Birational geometry via moduli spaces”, Birational geometry, rational curves, and arithmetic – Simons symposium, Springer-Verlag, New York, 2013, 93 – 132 · Zbl 1302.14035
[14] Yu. I. Manin, Frobenius manifolds, quantum cohomology, and moduli spaces, Amer. Math. Soc. Colloq. Publ., 47, Amer. Math. Soc., Providence, RI, 1999 · Zbl 0952.14032
[15] V. Golyshev, J. Stienstra, “Fuchsian equations of type DN”, Commun. Number Theory Phys., 1:2 (2007), 323 – 346 · Zbl 1158.34055
[16] В. В. Пржиялковский, “Минимальное кольцо Громова – Виттена”, Изв. РАН. Сер. матем., 72:6 (2008), 203 – 222 · Zbl 1158.53070
[17] V. V. Przyjalkowski, “Minimal Gromov – Witten rings”, Izv. Math., 72:6 (2008), 1253 – 1272 · Zbl 1158.53070
[18] V. V. Golyshev, “Classification problems and mirror duality”, Surveys in geometry and number theory: reports on contemporary Russian mathematics, London Math. Soc. Lecture Note Ser., 338, Cambridge Univ. Press, Cambridge, 2007, 88 – 121 · Zbl 1114.14024
[19] F. Beukers, J. Stienstra, “On the Picard – Fuchs equation and the formal Brauer group of certain elliptic K3-surfaces”, Math. Ann., 271:2 (1985), 269 – 304 · Zbl 0539.14006
[20] F. Beukers, “Irrationality of \(\pi^2\), periods of an elliptic curve and \(\Gamma^1(5)\)”, Diophantine approximations and transcendental numbers (Luminy, 1982), Progr. Math., 31, Birkhaüser, Boston, MA, 1984, 47 – 66 · Zbl 0518.10040
[21] В. И. Данилов, А. Г. Хованский, “Многогранники Ньютона и алгоритм вычисления чисел Ходжа – Делиня”, Изв. АН СССР. Сер. матем., 50:5 (1986), 925 – 945 · Zbl 0669.14012
[22] V. I. Danilov, A. G. Khovanskii\?, “Newton polyhedra and an algorithm for computing Hodge – Deligne numbers”, Math. USSR-Izv., 29:2 (1987), 279 – 298 · Zbl 0669.14012
[23] V. V. Batyrev, I. Ciocan-Fontanine, B. Kim, D. van Straten, “Conifold transitions and mirror symmetry for Calabi – Yau complete intersections in Grassmannians”, Nuclear Phys. B, 514:3 (1998), 640 – 666 · Zbl 0896.14025
[24] V. V. Batyrev, “Toric degenerations of Fano varieties and constructing mirror manifolds”, The Fano conference (Torino, Italy, 2002), Univ. di Torino, Torino, 2004, 109 – 122 · Zbl 1072.14070
[25] С. С. Галкин, Торические вырождения многообразий Фано, Дис. \cdots канд. физ.-матем. наук, МИАН, М., 2008
[27] V. V. Batyrev, “Dual polyhedra and mirror symmetry for Calabi – Yau hypersurfaces in toric varieties”, J. Algebraic Geom., 3:3 (1994), 493 – 535 · Zbl 0829.14023
[28] H. Hori, C. Vafa, Mirror symmetry, arXiv: hep-th/0002222
[29] V. Przyjalkowski, C. Shramov, On Hodge numbers of complete intersections and Landau – Ginzburg models, arXiv: 1305.4377 · Zbl 1319.14047
[30] В. В. Пржиялковский, “Квантовые когомологии гладких полных пересечений во взвешенных проективных пространствах и особых торических многообразиях”, Матем. сб., 198:9 (2007), 107 – 122 · Zbl 1207.14059
[31] V. V. Przyjalkowski, “Quantum cohomology of smooth complete intersections in weighted projective spaces and in singular toric varieties”, Sb. Math., 198:9 (2007), 1325 – 1340 · Zbl 1207.14059
[32] 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 · Zbl 1236.14038
[33] T. Eguchi, K. Hori, C.-Sh. Xiong, “Gravitational quantum cohomology”, Internat. J. Modern Phys. A, 12:9 (1997), 1743 – 1782 · Zbl 1072.32500
[34] A. Bertram, I. Ciocan-Fontanine, B. Kim, “Two proofs of a conjecture of Hori and Vafa”, Duke Math. J., 126:1 (2005), 101 – 136 · Zbl 1082.14055
[35] V. V. Batyrev, I. Ciocan-Fontanine, B. Kim, D. van Straten, “Mirror symmetry and toric degenerations of partial flag manifolds”, Acta Math., 184:1 (2000), 1 – 39 · Zbl 1022.14014
[36] A. Givental, “Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture”, Topics in singularity theory, Amer. Math. Soc. Transl. Ser. 2, 180, Amer. Math. Soc., Providence, RI, 1997, 103 – 115 · Zbl 0895.32006
[37] K. Rietsch, “A mirror symmetric solution to the quantum Toda lattice”, Comm. Math. Phys., 309:1 (2012), 23 – 49 · Zbl 1256.14042
[38] A. Bondal, S. Galkin, Mirror symmetry for minuscule varietics, IPMV 11-0101
[39] В. В. Пржиялковский, “Инварианты Громова – Виттена трехмерных многообразий Фано рода 6 и рода 8”, Матем. сб., 198:3 (2007), 145 – 158 · Zbl 1188.14038
[40] V. V. Przyjalkowski, “Gromov – Witten invariants of Fano threefolds of genera 6 and 8”, Sb. Math., 198:3 (2007), 433 – 446 · Zbl 1188.14038
[41] F. Beukers, C. A. M. Peters, “A family of K3 surfaces and \(\zeta(3)\)”, J. Reine Angew. Math., 351 (1984), 42 – 54 · Zbl 0541.14007
[42] A. Corti, V. Golyshev, “Hypergeometric equations and weighted projective spaces”, Sci. China Math., 54:8 (2011), 1577 – 1590 · Zbl 1237.14022
[43] A. Iliev, L. Katzarkov, V. Przyjalkowski, “Double solids, categories and non-rationality”, Proc. Edinb. Math. Soc. (2), Shokurov/s volume, 2013 (в печати) · Zbl 1303.14026
[44] arXiv: 1102.2130
[45] M. Gross, L. Katzarkov, H. Ruddat, “Towards mirror symmetry for varieties of general type”, J. Adv. Math. Stud (в печати) · Zbl 1371.14046
[46] arXiv: 1202.4042
[47] L. Katzarkov, V. Przyjalkowski, “Generalized homological mirror symmetry and cubics”, Многомерная алгебраическая геометрия, Тр. МИАН, 264, МАИК, М., 2009, 94 – 102 · Zbl 1312.14054
[48] L. Katzarkov, V. Przyjalkowski, “Generalized homological mirror symmetry and cubics”, Proc. Steklov Inst. Math., 264:1 (2009), 87 – 95 · Zbl 1312.14054
[49] T. Coates, A. Corti, S. Galkin, V. Golyshev, A. Kasprzyk, Fano varieties and extremal Laurent polynomials. A collaborative research blog, arXiv: · Zbl 1364.14032
[50] Ю. Г. Прохоров, “Степень трехмерных многообразий Фано с каноническими горенштейновыми особенностями”, Матем. сб., 196:1 (2005), 81 – 122 · Zbl 1081.14058
[51] Yu. G. Prokhorov, “On the degree of Fano threefolds with canonical Gorenstein singularities”, Sb. Math., 196:1 (2005), 77 – 114 · Zbl 1081.14058
[52] K. Altmann, “The versal deformation of an isolated toric Gorenstein singularity”, Invent. Math., 128:3 (1997), 443 – 479 · Zbl 0894.14025
[53] D. Auroux, L. Katzarkov, D. Orlov, “Mirror symmetry for weighted projective planes and their noncommutative deformations”, Ann. of Math. (2), 167:3 (2008), 867 – 943 · Zbl 1175.14030
[54] A. Kasprzyk, “Canonical toric Fano threefolds”, Canad. J. Math., 62:6 (2010), 1293 – 1309 · Zbl 1264.14055
[55] P. Hacking, Yu. Prokhorov, “Smoothable del Pezzo surfaces with quotient singularities”, Compos. Math., 146:1 (2010), 169 – 192 · Zbl 1194.14054
[56] S. Galkin, A. Usnich, Mutations of potentials, arXiv:
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