An analogue of Dubrovin’s conjecture

Sanda, Fumihiko; Shamoto, Yota

doi:10.5802/aif.3321

An analogue of Dubrovin’s conjecture
[Un analogue de la conjecture de Dubrovin]

Sanda, Fumihiko ¹ ; Shamoto, Yota ²

¹ Graduate school of Mathematics, Nagoya University, Furo-cho Chikusa-ku, Nagoya 464-8602 (Japan)
² Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo 5-1-5 Kashiwanoha Kashiwa, Chiba 277-8583 (Japan)

Annales de l'Institut Fourier, Tome 70 (2020) no. 2, pp. 621-682.

Résumé
Abstract

Nous proposons un analogue de la conjecture de Dubrovin pour le cas où les variétés de Fano ont des connexions quantiques de type exponentiel. Cela inclut le cas où les cohomologies quantiques ne sont pas nécessairement semi-simples. La conjecture est décrite comme un isomorphisme de structures algébriques linéaires, que nous appelons systèmes de mutation. Étant donné une telle variété de Fano $X$ , l’une des structures est donnée par la structure de Stokes de la connexion quantique de $X$ , et l’autre est donnée par une décomposition semi-orthogonale de la catégorie dérivées des faisceaux cohérents sur $X$ . De plus, nous prouvons la conjecture pour une classe d’intersections complètes lisses de Fano dans un espace projectif.

We propose an analogue of Dubrovin’s conjecture for the case where Fano manifolds have quantum connections of exponential type. It includes the case where the quantum cohomology rings are not necessarily semisimple. The conjecture is described as an isomorphism of two linear algebraic structures, which we call “mutation systems”. Given such a Fano manifold $X$ , one of the structures is given by the Stokes structure of the quantum connection of $X$ , and the other is given by a semiorthogonal decomposition of the derived category of coherent sheaves on $X$ . We also prove the conjecture for a class of smooth Fano complete intersections in a projective space.

Reçu le : 2017-09-11
Révisé le : 2018-11-15
Accepté le : 2019-01-17
Publié le : 2020-05-28

DOI : 10.5802/aif.3321

Classification : 14J33
Keywords: mirror symmetry, Fano manifolds, quantum cohomologies, Stokes matrix
Mot clés : symétrie miroir, variétés de Fano, cohomologie quantique, matrice de Stokes

Affiliations des auteurs :

Sanda, Fumihiko ¹ ; Shamoto, Yota ²

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Sanda, Fumihiko; Shamoto, Yota. An analogue of Dubrovin’s conjecture. Annales de l'Institut Fourier, Tome 70 (2020) no. 2, pp. 621-682. doi : 10.5802/aif.3321. https://aif.centre-mersenne.org/articles/10.5802/aif.3321/

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