Laurent phenomenon for Landau--Ginzburg models of complete intersections in Grassmannians

V. V. Przyjalkowski ; C. A. Shramov

Informatics and Automation, Modern problems of mathematics, mechanics, and mathematical physics, Tome 290 (2015), pp. 102-113.

Résumé

In 1997 Batyrev, Ciocan-Fontanine, Kim, and van Straten suggested a construction of Landau–Ginzburg models for Fano complete intersections in Grassmannians similar to Givental's construction for complete intersections in smooth toric varieties. We show that for a Fano complete intersection in a Grassmannian the result of the above construction is birational to a complex torus. In other words, the complete intersections under consideration have very weak Landau–Ginzburg models.

Bibliographie

[1] Aizenberg L.A., Tsikh A.K., Yuzhakov A.P., “Mnogomernye vychety i ikh prilozheniya”, Kompleksnyi analiz — mnogie peremennye–2, Itogi nauki i tekhniki. Sovr. probl. matematiki. Fund. napr., 8, VINITI, M., 1985, 5–64 | MR | Zbl

[2] Batyrev V.V., Ciocan-Fontanine I., Kim B., van Straten D., “Conifold transitions and mirror symmetry for Calabi–Yau complete intersections in Grassmannians”, Nucl. Phys. B, 514:3 (1998), 640–666 | DOI | MR | Zbl

[3] Batyrev V.V., Ciocan-Fontanine I., Kim B., van Straten D., “Mirror symmetry and toric degenerations of partial flag manifolds”, Acta math., 184:1 (2000), 1–39 | DOI | MR | Zbl

[4] Bertram A., Ciocan-Fontanine I., Kim B., “Two proofs of a conjecture of Hori and Vafa”, Duke Math. J., 126:1 (2005), 101–136 | DOI | MR | Zbl

[5] Coates T., Corti A., Galkin S., Golyshev V., Kasprzyk A., “Mirror symmetry and Fano manifolds”, European Congress of Mathematics, Proc. 6th ECM Congr., Kraków, July 2–7, 2012, Eur. Math. Soc., Zürich, 2013, 285–300

[6] Doran C.F., Harder A., Toric degenerations and the Laurent polynomials related to Givental's Landau–Ginzburg models, E-print, 2015, arXiv: 1502.02079 [math.AG]

[7] Eguchi T., Hori K., Xiong C.-S., “Gravitational quantum cohomology”, Int. J. Mod. Phys. A, 12:9 (1997), 1743–1782 | DOI | MR | Zbl

[8] Givental A.B., “Equivariant Gromov–Witten invariants”, Int. Math. Res. Not., 1996:13 (1996), 613–663 | DOI | MR | Zbl

[9] Golyshev V., Stienstra J., “Fuchsian equations of type DN”, Commun. Number Theory Phys., 1:2 (2007), 323–346 | DOI | MR | Zbl

[10] Ilten N.O., Lewis J., Przyjalkowski V., “Toric degenerations of Fano threefolds giving weak Landau–Ginzburg models”, J. Algebra, 374 (2013), 104–121 | DOI | MR | Zbl

[11] Manin Yu.I., Frobeniusovy mnogoobraziya, kvantovye kogomologii, prostranstva modulei, Faktorial, M., 2002

[12] Marsh R.J., Rietsch K., The $B$-model connection and mirror symmetry for Grassmannians, E-print, 2013, arXiv: 1307.1085 [math.AG]

[13] Pandharipande R., “Rational curves on hypersurfaces [after A. Givental]”, Séminaire Bourbaki. Volume 1997/98. Exposés 835–849, Exp. 848, Astérisque, 252, Soc. math. France, Paris, 1998, 307–340 | MR | Zbl

[14] Przyjalkowski V., “On Landau–Ginzburg models for Fano varieties”, Commun. Number Theory Phys., 1:4 (2008), 713–728 | DOI | MR | Zbl

[15] Przhiyalkovskii V.V., “Minimalnoe koltso Gromova–Vittena”, Izv. RAN. Ser. mat., 72:6 (2008), 203–222 | DOI | MR

[16] Przyjalkowski V., “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 | DOI | MR | Zbl

[17] Przhiyalkovskii V.V., “Slabye modeli Landau–Ginzburga gladkikh trekhmernykh mnogoobrazii Fano”, Izv. RAN. Ser. mat., 77:4 (2013), 135–160 | DOI | MR

[18] Przyjalkowski V., Shramov C., Laurent phenomenon for Landau–Ginzburg models of complete intersections in Grassmannians of planes, E-print, 2014, arXiv: 1409.3729 [math.AG]

[19] Przhiyalkovskii V.V., Shramov K.A., “O slabykh modelyakh Landau–Ginzburga dlya polnykh peresechenii v grassmanianakh”, UMN, 69:6 (2014), 181–182 | DOI

[20] Przyjalkowski V., Shramov C., “On Hodge numbers of complete intersections and Landau–Ginzburg models”, Int. Math. Res. Not., 2015 | DOI