Narrow quantum D-modules and quantum Serre duality
[D-modules et dualité de Serre quantiques dans le secteur troit]
Annales de l'Institut Fourier, Online first, 49 p.

Étant donnée une variété ou une orbifold non compacte 𝒴, on définit un sous-espace naturel de la cohomologie de 𝒴 que nous appelons cohomologie étroite. On montre qu’en dépit du fait que 𝒴 est non compacte, il existe un couplage non-dégénéré sur ce sous-espace. Cette cohomologie étroite s’avère utile pour l’étude de la théorie de Gromov–Witten en genre 0. Lorsque 𝒴 est une variété complexe lisse ou un champ de Deligne–Mumford lisse, on peut définir un D-module quantique sur sa cohomologie étroite. Ceci nous amène à une nouvelle formulation de la dualité de Serre quantique.

Given 𝒴 a non-compact manifold or orbifold, we define a natural subspace of the cohomology of 𝒴 called the narrow cohomology. We show that despite 𝒴 being non-compact, there is a well-defined and non-degenerate pairing on this subspace. The narrow cohomology proves useful for the study of genus zero Gromov–Witten theory. When 𝒴 is a smooth complex variety or Deligne–Mumford stack, one can define a quantum D-module on the narrow cohomology of 𝒴. This yields a new formulation of quantum Serre duality.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : https://doi.org/10.5802/aif.3419
Classification : 14N35,  53D45,  14E16
Mots clés : théorie de Gromov–Witten, D-modules quantiques, dualité de Serre quantique, symétrie miroir
@unpublished{AIF_0__0_0_A18_0,
     author = {Shoemaker, Mark},
     title = {Narrow quantum $D$-modules and quantum {Serre} duality},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2021},
     doi = {10.5802/aif.3419},
     language = {en},
     note = {Online first},
}
Shoemaker, Mark. Narrow quantum $D$-modules and quantum Serre duality. Annales de l'Institut Fourier, Online first, 49 p.

[1] Abramovich, Dan; Graber, Tom; Vistoli, Angelo Gromov-Witten theory of Deligne-Mumford stacks, Am. J. Math., Volume 130 (2008) no. 5, pp. 1337-1398 | Article

[2] Abramovich, Dan; Vistoli, Angelo Compactifying the space of stable maps, J. Am. Math. Soc., Volume 15 (2002) no. 1, pp. 27-75 | Article | MR 1862797

[3] Adem, Alejandro; Leida, Johann; Ruan, Yongbin Orbifolds and stringy topology, Cambridge Tracts in Mathematics, 171, Cambridge University Press, 2007, xii+149 pages | Article | MR 2359514

[4] Atiyah, Michael F.; Bott, Raoul; Shapiro, Arnold Clifford modules, Topology, Volume 3 (1964) no. suppl. 1, pp. 3-38 | Article | MR 0167985 | Zbl 0146.19001

[5] Ballard, Matthew; Favero, David; Katzarkov, Ludmil A category of kernels for equivariant factorizations and its implications for Hodge theory, Publ. Math., Inst. Hautes Étud. Sci., Volume 120 (2014), pp. 1-111 | Article | MR 3270588

[6] Behrend, Kai; Fantechi, Barbara The intrinsic normal cone, Invent. Math., Volume 128 (1997) no. 1, pp. 45-88 | Article | MR 1437495

[7] Borisov, Lev A.; Horja, R. Paul Applications of homological mirror symmetry to hypergeometric systems: duality conjectures, Adv. Math., Volume 271 (2015), pp. 153-187 | Article | MR 3291860

[8] Bott, Raoul; Tu, Loring W. Differential forms in algebraic topology, Graduate Texts in Mathematics, 82, Springer, 1982, xiv+331 pages | MR 658304

[9] Bredon, Glen E. Sheaf theory, Graduate Texts in Mathematics, 170, Springer, 1997, xii+502 pages | Article | MR 1481706

[10] Chen, Weimin; Ruan, Yongbin A new cohomology theory of orbifold, Commun. Math. Phys., Volume 248 (2004) no. 1, pp. 1-31 | Article

[11] Chiodo, Alessandro; Iritani, Hiroshi; Ruan, Yongbin Landau–Ginzburg/Calabi–Yau correspondence, global mirror symmetry and Orlov equivalence, Publ. Math., Inst. Hautes Étud. Sci., Volume 119 (2014), pp. 127-216 | Article | MR 3210178

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

[13] Coates, Tom; Gholampour, Amin; Iritani, Hiroshi; Jiang, Yunfeng; Johnson, Paul; Manolache, Cristina The quantum Lefschetz hyperplane principle can fail for positive orbifold hypersurfaces, Math. Res. Lett., Volume 19 (2012) no. 5, pp. 997-1005 | Article | MR 3039825

[14] Coates, Tom; Givental, Alexander B. Quantum Riemann–Roch, Lefschetz and Serre, Ann. Math., Volume 165 (2007) no. 1, pp. 15-53 | Article | MR 2276766

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

[16] Fulton, William Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 2, Springer, 1984, xi+470 pages | Article | MR 732620

[17] Givental, Alexander B. Equivariant Gromov–Witten invariants, Int. Math. Res. Not. (1996) no. 13, pp. 613-663 | Article

[18] Givental, Alexander B. Elliptic Gromov–Witten invariants and the generalized mirror conjecture, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), World Scientific, 1998, pp. 107-155 | MR 1672116

[19] Givental, Alexander B. Symplectic geometry of Frobenius structures, Frobenius manifolds (Aspects of Mathematics), Volume E36, Friedr. Vieweg, 2004, pp. 91-112

[20] Graber, Tom; Pandharipande, Rahul Localization of virtual classes, Invent. Math., Volume 135 (1999) no. 2, pp. 487-518 | Article

[21] Iritani, Hiroshi An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math., Volume 222 (2009) no. 3, pp. 1016-1079 | Article | MR 2553377

[22] Iritani, Hiroshi Ruan’s conjecture and integral structures in quantum cohomology, New developments in algebraic geometry, integrable systems and mirror symmetry (RIMS, Kyoto, 2008) (Advanced Studies in Pure Mathematics), Volume 59, Mathematical Society of Japan, 2010, pp. 111-166 | MR 2683208

[23] Iritani, Hiroshi Quantum cohomology and periods, Ann. Inst. Fourier, Volume 61 (2011) no. 7, pp. 2909-2958 | MR 3112512

[24] Iritani, Hiroshi; Mann, Etienne; Mignon, Thierry Quantum Serre theorem as a duality between quantum D-modules, Int. Math. Res. Not., Volume 2016 (2016) no. 9, pp. 2828-2888 | Article | MR 3519131

[25] Iversen, Birger Local Chern classes, Ann. Sci. Éc. Norm. Supér., Volume 9 (1976) no. 1, pp. 155-169 | MR 0460332

[26] Lee, Yuan-Pin; Priddis, Nathan; Shoemaker, Mark A proof of the Landau–Ginzburg/Calabi–Yau correspondence via the crepant transformation conjecture, Ann. Sci. Éc. Norm. Supér., Volume 49 (2016) no. 6, pp. 1403-1443 | Article | MR 3592361

[27] Mi, Rongxiao; Shoemaker, Mark Extremal transitions via quantum Serre duality (2020) (https://arxiv.org/abs/2006.09907)

[28] Pandharipande, Rahul Rational curves on hypersurfaces (after A. Givental), Séminaire Bourbaki. Vol. 1997/98 (Astérisque), Volume 252, Société Mathématique de France, 1998, pp. 307-340 | MR 1685628

[29] Shoemaker, Mark Integral transforms and quantum correspondences, Adv. Math., Volume 374 (2020), 107381, 59 pages

[30] Toën, Bertrand Théorèmes de Riemann–Roch pour les champs de Deligne–Mumford, K-Theory, Volume 18 (1999) no. 1, pp. 33-76 | Article | MR 1710187

[31] Tseng, Hsian-Hua Orbifold quantum Riemann–Roch, Lefschetz and Serre, Geom. Topol., Volume 14 (2010) no. 1, pp. 1-81 | Article