no. 3
On quantum cohomology of Grassmannians of isotropic lines, unfoldings of A n -singularities, and Lefschetz exceptional collections
[Sur la cohomologie quantique des grassmanniennes de droites isotropes, le déploiement de singularités de type A n et les collections de Lefschetz exceptionnelles.]
Cruz Morales, John Alexander1 ; Mellit, Anton2 ; Perrin, Nicolas3 ; Smirnov, Maxim4
1 Universidad Nacional de Colombia Departamento de Matemáticas Carrera 45 No. 26–85 Edificio Uriel Gutiérrez Bogotá D.C. (Colombia)
2 Faculty of Mathematics University of Vienna Oskar-Morgenstern-Platz 1 1090 Vienna (Austria)
3 Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris–Saclay 78035 Versailles (France)
4 Universität Augsburg, Institut für Mathematik Universitätsstr. 14 86159 Augsburg (Germany)
Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 955-991.

Dans cet article, nous nous intéressons au gros anneau de cohomologie quantique de IG(2,2n), la grassmanienne symplectique des droites isotropes. Nous montrons que cet anneau est régulier et en déduisons par « lissité générique » une preuve conceptuelle de la semi-simplicité générique du gros anneau de cohomologie quantique de IG(2,2n). Par ailleurs, par un résultat général de Hertling, cette régularité donne une description de cet anneau en termes de singularités isolées d’hypersurfaces et nous montrons que les singularités qui apparaissent sont de type A n-1 . La conjecture de symétrie miroir homologique prédit l’existence de suites exceptionnelles très spéciales dans la catégorie dérivée des faisceaux cohérents de IG(2,2n). L’existence de telles collections est démontrée en appendice par Alexander Kuznetsov.

The subject of this paper is the big quantum cohomology rings of symplectic isotropic Grassmannians IG(2,2n). We show that these rings are regular. In particular, by “generic smoothness”, we obtain a conceptual proof of generic semisimplicity of the big quantum cohomology for IG(2,2n). Further, by a general result of Hertling, the regularity of these rings implies that they have a description in terms of isolated hypersurface singularities, which we show in this case to be of type A n-1 . By the homological mirror symmetry conjecture, these results suggest the existence of a very special full exceptional collection in the derived category of coherent sheaves on IG(2,2n). Such a collection is constructed in the appendix by Alexander Kuznetsov.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3263
Classification : 14N35, 53D45
Keywords: semisimplicity of quantum cohomology, unfoldings of singularities, Lefschetz exceptional collections
Mot clés : semi-simplicité de la cohomologie quantique, déploiement des singularités, collections de Lefschetz exceptionnelles

Cruz Morales, John Alexander 1 ; Mellit, Anton 2 ; Perrin, Nicolas 3 ; Smirnov, Maxim 4

1 Universidad Nacional de Colombia Departamento de Matemáticas Carrera 45 No. 26–85 Edificio Uriel Gutiérrez Bogotá D.C. (Colombia)
2 Faculty of Mathematics University of Vienna Oskar-Morgenstern-Platz 1 1090 Vienna (Austria)
3 Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris–Saclay 78035 Versailles (France)
4 Universität Augsburg, Institut für Mathematik Universitätsstr. 14 86159 Augsburg (Germany)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2019__69_3_955_0,
     author = {Cruz Morales, John Alexander and Mellit, Anton and Perrin, Nicolas and Smirnov, Maxim},
     title = {On quantum cohomology of {Grassmannians} of isotropic lines, unfoldings of $A_n$-singularities, and {Lefschetz} exceptional collections},
     journal = {Annales de l'Institut Fourier},
     pages = {955--991},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {3},
     year = {2019},
     doi = {10.5802/aif.3263},
     zbl = {07067424},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3263/}
}
TY  - JOUR
AU  - Cruz Morales, John Alexander
AU  - Mellit, Anton
AU  - Perrin, Nicolas
AU  - Smirnov, Maxim
TI  - On quantum cohomology of Grassmannians of isotropic lines, unfoldings of $A_n$-singularities, and Lefschetz exceptional collections
JO  - Annales de l'Institut Fourier
PY  - 2019
SP  - 955
EP  - 991
VL  - 69
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3263/
DO  - 10.5802/aif.3263
LA  - en
ID  - AIF_2019__69_3_955_0
ER  - 
%0 Journal Article
%A Cruz Morales, John Alexander
%A Mellit, Anton
%A Perrin, Nicolas
%A Smirnov, Maxim
%T On quantum cohomology of Grassmannians of isotropic lines, unfoldings of $A_n$-singularities, and Lefschetz exceptional collections
%J Annales de l'Institut Fourier
%D 2019
%P 955-991
%V 69
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3263/
%R 10.5802/aif.3263
%G en
%F AIF_2019__69_3_955_0
Cruz Morales, John Alexander; Mellit, Anton; Perrin, Nicolas; Smirnov, Maxim. On quantum cohomology of Grassmannians of isotropic lines, unfoldings of $A_n$-singularities, and Lefschetz exceptional collections. Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 955-991. doi : 10.5802/aif.3263. https://aif.centre-mersenne.org/articles/10.5802/aif.3263/

[1] Arnold, Vladimir I.; Gusein-Zade, Sabir M.; Varchenko, Alexander N. Singularities of differentiable maps. Volume 1. Classification of critical points, caustics and wave fronts, Modern Birkhäuser Classics, Birkhäuser, 2012, xii+382 pages (translated from the Russian by Ian Porteous based on a previous translation by Mark Reynolds, reprint of the 1985 edition) | MR | Zbl

[2] Bayer, Arend Semisimple quantum cohomology and blowups, Int. Math. Res. Not. (2004) no. 40, pp. 2069-2083 | DOI | MR | Zbl

[3] Buch, Anders Skovsted; Kresch, Andrew; Tamvakis, Harry Quantum Pieri rules for isotropic Grassmannians, Invent. Math., Volume 178 (2009) no. 2, pp. 345-405 | DOI | MR | Zbl

[4] Chaput, Pierre-Emmanuel; Manivel, Laurent; Perrin, Nicolas Quantum cohomology of minuscule homogeneous spaces III. Semi-simplicity and consequences, Can. J. Math., Volume 62 (2010) no. 6, pp. 1246-1263 | DOI | MR | Zbl

[5] Chaput, Pierre-Emmanuel; Perrin, Nicolas On the quantum cohomology of adjoint varieties, Proc. Lond. Math. Soc., Volume 103 (2011) no. 2, pp. 294-330 | DOI | MR | Zbl

[6] Coates, Tom; Iritani, Hiroshi On the convergence of Gromov-Witten potentials and Givental’s formula, Mich. Math. J., Volume 64 (2015) no. 3, pp. 587-631 | DOI | MR | Zbl

[7] Dubrovin, Boris Geometry and analytic theory of Frobenius manifolds, Doc. Math. (1998), pp. 315-326 Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998) | MR | Zbl

[8] Dubrovin, Boris Quantum cohomology and isomonodromic deformation (2013) (Lecture at “Recent Progress in the Theory of Painleve Equations: Algebraic, asymptotic and topological aspects", Strasbourg,)

[9] Fonarëv, A. V. Minimal Lefschetz decompositions of the derived categories for Grassmannians, Izv. Math., Volume 77 (2013) no. 5, pp. 203-224 | MR | Zbl

[10] Fulton, William; Pandharipande, Rahul Notes on stable maps and quantum cohomology, Algebraic geometry—Santa Cruz 1995 (Proceedings of Symposia in Pure Mathematics), Volume 62, American Mathematical Society, 1997, pp. 45-96 | DOI | MR | Zbl

[11] Galkin, Sergey; Golyshev, Vasily; Iritani, Hiroshi Gamma classes and quantum cohomology of Fano manifolds: gamma conjectures, Duke Math. J., Volume 165 (2016) no. 11, pp. 2005-2077 | DOI | MR | Zbl

[12] Galkin, Sergey; Mellit, Anton; Smirnov, Maxim Dubrovin’s conjecture for IG (2,6), Int. Math. Res. Not. (2015) no. 18, pp. 8847-8859 | DOI | MR | Zbl

[13] Ganatra, Sheel Automatically generating Fukaya categories and computing quantum cohomology (2016) (https://arxiv.org/abs/1605.07702)

[14] Hertling, Claus (Email communication dated 04.04.2016) | Numdam

[15] Hertling, Claus; Manin, Yuri I Weak Frobenius manifolds, Int. Math. Res. Not. (1999) no. 6, pp. 277-286 | DOI | MR | Zbl

[16] Hertling, Claus; Manin, Yuri I; Teleman, Constatin An update on semisimple quantum cohomology and F-manifolds, Tr. Mat. Inst. Steklova, Volume 264 (2009), pp. 69-76 | DOI | MR | Zbl

[17] Hertling, Claus Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts in Mathematics, 151, Cambridge University Press, 2002, x+270 pages | DOI | MR | Zbl

[18] Kleiman, Steven L. The transversality of a general translate, Compos. Math., Volume 28 (1974), pp. 287-297 | MR | Zbl

[19] Kuznetsov, Alexander Homological projective duality, Publ. Math., Inst. Hautes �tud. Sci. (2007) no. 105, pp. 157-220 | DOI | MR | Zbl

[20] Kuznetsov, Alexander Exceptional collections for Grassmannians of isotropic lines, Proc. Lond. Math. Soc., Volume 97 (2008) no. 1, pp. 155-182 | DOI | MR | Zbl

[21] Kuznetsov, Alexander Semiorthogonal decompositions in algebraic geometry, Proceedings of the International Congress of Mathematicians, Vol. II (Seoul, 2014) (2014), pp. 635-660 | MR | Zbl

[22] Landsberg, Joseph M.; Manivel, Laurent On the projective geometry of rational homogeneous varieties, Comment. Math. Helv., Volume 78 (2003) no. 1, pp. 65-100 | DOI | MR | Zbl

[23] Lin, Kevin H.; Pomerleano, Daniel Global matrix factorizations, Math. Res. Lett., Volume 20 (2013) no. 1, pp. 91-106 | DOI | MR | Zbl

[24] Manin, Yuri I. Frobenius manifolds, quantum cohomology, and moduli spaces, Colloquium Publications, 47, American Mathematical Society, 1999, xiv+303 pages | DOI | MR | Zbl

[25] Orlov, Dmitri Landau-Ginzburg models, D-branes and mirror symmetry, Mat. Contemp., Volume 41 (2012), pp. 75-112 | MR | Zbl

[26] Perrin, Nicolas Semisimple quantum cohomology of some Fano varieties (2014) (https://arxiv.org/abs/1405.5914v1)

[27] Ruan, Yongbin; Tian, Gang A mathematical theory of quantum cohomology, J. Differ. Geom., Volume 42 (1995) no. 2, pp. 259-367 | DOI | MR | Zbl

[28] Samokhin, Alexander Some remarks on the derived categories of coherent sheaves on homogeneous spaces, J. Lond. Math. Soc., Volume 76 (2007) no. 1, pp. 122-134 | DOI | MR | Zbl

[29] Seidel, Paul More about vanishing cycles and mutation, Symplectic geometry and mirror symmetry (Seoul, 2000), World Scientific, 2001, pp. 429-465 | DOI | MR | Zbl

[30] Stanley, Richard P. Hilbert functions of graded algebras, Adv. Math., Volume 28 (1978) no. 1, pp. 57-83 | DOI | MR | Zbl

[31] Strickland, Elisabetta Lines in G/P, Math. Z., Volume 242 (2002) no. 2, pp. 227-240 | DOI | MR | Zbl

Cité par Sources :