On quantum cohomology of Grassmannians of isotropic lines, unfoldings of A n -singularities, and Lefschetz exceptional collections
Annales de l'Institut Fourier, Volume 69 (2019) no. 3, pp. 955-991.

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.

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.

Published online:
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)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {On quantum cohomology of {Grassmannians} of isotropic lines, unfoldings of $A_n$-singularities, and {Lefschetz} exceptional collections},
     journal = {Annales de l'Institut Fourier},
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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, Volume 69 (2019) no. 3, pp. 955-991. doi : 10.5802/aif.3263. https://aif.centre-mersenne.org/articles/10.5802/aif.3263/

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