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

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 :
Révisé le :
Accepté le :
Publié le :
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
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)
@article{AIF_2020__70_2_621_0,
     author = {Sanda, Fumihiko and Shamoto, Yota},
     title = {An analogue of {Dubrovin{\textquoteright}s} conjecture},
     journal = {Annales de l'Institut Fourier},
     pages = {621--682},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {70},
     number = {2},
     year = {2020},
     doi = {10.5802/aif.3321},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3321/}
}
TY  - JOUR
AU  - Sanda, Fumihiko
AU  - Shamoto, Yota
TI  - An analogue of Dubrovin’s conjecture
JO  - Annales de l'Institut Fourier
PY  - 2020
SP  - 621
EP  - 682
VL  - 70
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.3321/
DO  - 10.5802/aif.3321
LA  - en
ID  - AIF_2020__70_2_621_0
ER  - 
%0 Journal Article
%A Sanda, Fumihiko
%A Shamoto, Yota
%T An analogue of Dubrovin’s conjecture
%J Annales de l'Institut Fourier
%D 2020
%P 621-682
%V 70
%N 2
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.3321/
%R 10.5802/aif.3321
%G en
%F AIF_2020__70_2_621_0
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. http://archive.numdam.org/articles/10.5802/aif.3321/

[1] Balser, Werner; Jurkat, Wolfgang B.; Lutz, Donald A. Birkhoff invariants and Stokes’ multipliers for meromorphic linear differential equations, J. Math. Anal. Appl., Volume 71 (1979) no. 1, pp. 48-94 | DOI | MR | Zbl

[2] Bayer, Arend; Manin, Yuri I. (Semi)simple exercises in quantum cohomology, The Fano Conference (Torino, 2002), Università di Torino, 2004, pp. 143-173 | MR | Zbl

[3] Bondal, Alexey I. A symplectic groupoid of triangular bilinear forms and the braid group, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 68 (2004) no. 4, pp. 19-74 | DOI | MR | Zbl

[4] Bondal, Alexey I.; Kapranov, Mikhail M. Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 53 (1989) no. 6, p. 1183-1205, 1337 | MR | Zbl

[5] Bourbaki, Nicolas Lie groups and Lie algebras. Chapters 4–6, Springer, 2002, xii+300 pages (Translated from the 1968 French original by Andrew Pressley) | DOI | MR | Zbl

[6] Căldăraru, Andrei The Mukai pairing. II. The Hochschild–Kostant–Rosenberg isomorphism, Adv. Math., Volume 194 (2005) no. 1, pp. 34-66 | DOI | MR | Zbl

[7] Căldăraru, Andrei; Willerton, Simon The Mukai pairing. I. A categorical approach, New York J. Math., Volume 16 (2010), pp. 61-98 | MR | Zbl

[8] Canonaco, Alberto; Stellari, Paolo Non-uniqueness of Fourier–Mukai kernels, Math. Z., Volume 272 (2012) no. 1-2, pp. 577-588 | DOI | MR | Zbl

[9] 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

[10] Ciolli, Gianni On the quantum cohomology of some Fano threefolds and a conjecture of Dubrovin, Int. J. Math., Volume 16 (2005) no. 8, pp. 823-839 | DOI | MR | Zbl

[11] Coates, Tom The Quantum Lefschetz Principle for Vector Bundles as a Map Between Givental Cones (2014) (https://arxiv.org/abs/1405.2893)

[12] Cox, David A.; Katz, Sheldon Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, 68, American Mathematical Society, 1999, xxii+469 pages | DOI | MR | Zbl

[13] Cruz Morales, John A.; van der Put, Marius Stokes matrices for the quantum differential equations of some Fano varieties, Eur. J. Math., Volume 1 (2015) no. 1, pp. 138-153 | DOI | MR | Zbl

[14] Dubrovin, Boris Geometry of 2D topological field theories, Integrable systems and quantum groups (Montecatini Terme, 1993) (Lecture Notes in Mathematics), Volume 1620, Springer, 1996, pp. 120-348 | DOI | MR | Zbl

[15] Dubrovin, Boris Geometry and analytic theory of Frobenius manifolds, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998) (1998) no. Extra Vol. II, pp. 315-326 | MR | Zbl

[16] Dubrovin, Boris Painlevé transcendents in two-dimensional topological field theory, The Painlevé property (CRM Series in Mathematical Physics), Springer, 1999, pp. 287-412 | DOI | MR | Zbl

[17] 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

[18] Galkin, Sergey; Iritani, Hiroshi Gamma conjecture via mirror symmetry (2015) (https://arxiv.org/abs/1508.00719)

[19] 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

[20] Givental, Alexander Equivariant Gromov–Witten invariants, Int. Math. Res. Not. (1996) no. 13, pp. 613-663 | DOI | MR | Zbl

[21] Guzzetti, Davide Stokes matrices and monodromy of the quantum cohomology of projective spaces, Commun. Math. Phys., Volume 207 (1999) no. 2, pp. 341-383 | DOI | MR | Zbl

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

[23] Hertling, Claus; Sabbah, Claude Examples of non-commutative Hodge structures, J. Inst. Math. Jussieu, Volume 10 (2011) no. 3, pp. 635-674 | DOI | MR | Zbl

[24] Hochschild, Gerhard; Kostant, Bertram; Rosenberg, Alex Differential forms on regular affine algebras, Trans. Am. Math. Soc., Volume 102 (1962), pp. 383-408 | DOI | MR | Zbl

[25] Huybrechts, Daniel Fourier–Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, Clarendon Press, 2006, viii+307 pages | DOI | MR | Zbl

[26] Iritani, Hiroshi An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math., Volume 222 (2009) no. 3, pp. 1016-1079 | DOI | MR | Zbl

[27] Iritani, Hiroshi Quantum cohomology and periods, Ann. Inst. Fourier, Volume 61 (2011) no. 7, pp. 2909-2958 | DOI | Numdam | MR | Zbl

[28] Iritani, Hiroshi; Mann, Etienne; Mignon, Thierry Quantum Serre theorem as a duality between quantum D-modules, Int. Math. Res. Not. (2016) no. 9, pp. 2828-2888 | DOI | MR | Zbl

[29] Iwaki, Kohei; Takahashi, Atsushi Stokes matrices for the quantum cohomologies of a class of orbifold projective lines, J. Math. Phys., Volume 54 (2013) no. 10, 101701, 18 pages | DOI | MR | Zbl

[30] Katzarkov, Ludmil; Kontsevich, Maxim; Pantev, Tony Hodge theoretic aspects of mirror symmetry, From Hodge theory to integrability and TQFT tt*-geometry (Proceedings of Symposia in Pure Mathematics), Volume 78, American Mathematical Society, 2008, pp. 87-174 | DOI | MR | Zbl

[31] Katzarkov, Ludmil; Kontsevich, Maxim; Pantev, Tony Bogomolov–Tian–Todorov theorems for Landau–Ginzburg models, J. Differ. Geom., Volume 105 (2017) no. 1, pp. 55-117 | DOI | MR | Zbl

[32] Kuznetsov, Alexander Hochschild homology and semiorthogonal decompositions (2009) (https://arxiv.org/abs/0904.4330)

[33] Kuznetsov, Alexander Base change for semiorthogonal decompositions, Compos. Math., Volume 147 (2011) no. 3, pp. 852-876 | DOI | MR | Zbl

[34] Kuznetsov, Alexander Exceptional collections in surface-like categories, Mat. Sb., Volume 208 (2017) no. 9, pp. 116-147 | DOI | MR | Zbl

[35] Lazarsfeld, Robert Positivity in algebraic geometry. II. Positivity for Vector Bundles, and Multiplier Ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 49, Springer, 2004, xviii+385 pages | DOI | MR | Zbl

[36] Libgober, Anatoly Chern classes and the periods of mirrors, Math. Res. Lett., Volume 6 (1999) no. 2, pp. 141-149 | DOI | MR | Zbl

[37] Lunts, Valery A. Lefschetz fixed point theorems for Fourier-Mukai functors and DG algebras, J. Algebra, Volume 356 (2012), pp. 230-256 | DOI | MR | Zbl

[38] Malgrange, Bernard La classification des connexions irrégulières à une variable, Mathematics and physics (Paris, 1979/1982) (Progress in Mathematics), Volume 37, Birkhäuser, 1983, pp. 381-399 | MR | Zbl

[39] Mann, Etienne; Mignon, Thierry Quantum 𝒟-modules for toric nef complete intersections, Int. J. Math., Volume 28 (2017) no. 6, 1750047, 49 pages | DOI | MR | Zbl

[40] Marcolli, Matilde; Tabuada, Gonçalo From exceptional collections to motivic decompositions via noncommutative motives, J. Reine Angew. Math., Volume 701 (2015), pp. 153-167 | DOI | MR | Zbl

[41] Matsumoto, Hideya Générateurs et relations des groupes de Weyl généralisés, C. R. Math. Acad. Sci. Paris, Volume 258 (1964), pp. 3419-3422 | MR | Zbl

[42] Mukai, Shigeru Polarized K3 surfaces of genus 18 and 20, Complex projective geometry (Trieste, 1989/Bergen, 1989) (London Mathematical Society Lecture Note Series), Volume 179, Cambridge University Press, 1992, pp. 264-276 | DOI | MR | Zbl

[43] Nogin, Dmitry Yu. Helices on some Fano threefolds: constructivity of semiorthogonal bases of K 0 , Ann. Sci. Éc. Norm. Supér., Volume 27 (1994) no. 2, pp. 129-172 | DOI | Numdam | MR | Zbl

[44] Polishchuk, Alexander Lefschetz type formulas for dg-categories, Sel. Math., New Ser., Volume 20 (2014) no. 3, pp. 885-928 | DOI | MR | Zbl

[45] Polishchuk, Alexander; Vaintrob, Arkady Chern characters and Hirzebruch–Riemann–Roch formula for matrix factorizations, Duke Math. J., Volume 161 (2012) no. 10, pp. 1863-1926 | DOI | MR | Zbl

[46] Ramadoss, Ajay C. The relative Riemann-Roch theorem from Hochschild homology, New York J. Math., Volume 14 (2008), pp. 643-717 | MR | Zbl

[47] Ramadoss, Ajay C. The Mukai pairing and integral transforms in Hochschild homology, Mosc. Math. J., Volume 10 (2010) no. 3, pp. 629-645 | DOI | MR | Zbl

[48] Reichelt, Thomas; Sevenheck, Christian Non-affine Landau–Ginzburg models and intersection cohomology, Ann. Sci. Éc. Norm. Supér., Volume 50 (2017) no. 3, pp. 665-753 | DOI | MR | Zbl

[49] Sabbah, Claude Isomonodromic deformations and Frobenius manifolds. An introduction, Universitext, Springer; EDP Sciences, 2007, xiv+279 pages | MR | Zbl

[50] Sabbah, Claude Non-commutative Hodge structures, Ann. Inst. Fourier, Volume 61 (2011) no. 7, pp. 2681-2717 | DOI | Numdam | MR | Zbl

[51] Sabbah, Claude Introduction to Stokes structures, Lecture Notes in Mathematics, 2060, Springer, 2013, xiv+249 pages | DOI | MR | Zbl

[52] Shklyarov, D. Hirzebruch–Riemann–Roch-type formula for DG algebras, Proc. Lond. Math. Soc., Volume 106 (2013) no. 1, pp. 1-32 | DOI | MR | Zbl

[53] Swan, Richard G. Hochschild cohomology of quasiprojective schemes, J. Pure Appl. Algebra, Volume 110 (1996) no. 1, pp. 57-80 | DOI | MR | Zbl

[54] Ueda, Kazushi Stokes matrices for the quantum cohomologies of Grassmannians, Int. Math. Res. Not. (2005) no. 34, pp. 2075-2086 | DOI | MR | Zbl

[55] Ueda, Kazushi Stokes Matrix for the Quantum Cohomology of Cubic Surfaces (2005) (https://arxiv.org/abs/math/0505350) | Zbl

[56] de Thanhoffer de Volcsey, Louis; van den Bergh, Michel On an analogue of the Markov equation for exceptional collections of length 4 (2016) (https://arxiv.org/abs/1607.04246)

[57] Yekutieli, Amnon The continuous Hochschild cochain complex of a scheme, Can. J. Math., Volume 54 (2002) no. 6, pp. 1319-1337 | DOI | MR | Zbl

Cité par Sources :