An analogue of Dubrovin’s conjecture
Annales de l'Institut Fourier, Volume 70 (2020) no. 2, pp. 621-682.

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.

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.

Received: 2017-09-11
Revised: 2018-11-15
Accepted: 2019-01-17
Published online: 2020-05-28
DOI: https://doi.org/10.5802/aif.3321
Classification: 14J33
Keywords: mirror symmetry, Fano manifolds, quantum cohomologies, Stokes matrix
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     author = {Sanda, Fumihiko and Shamoto, Yota},
     title = {An analogue of Dubrovin's conjecture},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {70},
     number = {2},
     year = {2020},
     pages = {621-682},
     doi = {10.5802/aif.3321},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2020__70_2_621_0/}
}
Sanda, Fumihiko; Shamoto, Yota. An analogue of Dubrovin’s conjecture. Annales de l'Institut Fourier, Volume 70 (2020) no. 2, pp. 621-682. doi : 10.5802/aif.3321. https://aif.centre-mersenne.org/item/AIF_2020__70_2_621_0/

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