A proof of the Landau-Ginzburg/ Calabi-Yau correspondence via the crepant transformation conjecture
[Une preuve de la correspondance de Landau-Ginzburg/Calabi-Yau via la conjecture de la transformation crépante]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 6, pp. 1403-1443.

Nous établissons une nouvelle relation (la correspondance MLK) entre la théorie FJRW twistée et la théorie de Gromov-Witten en tout genre. Cela nous permet de montrer que la conjecture de la transformation crépante pour le type de Fermat en genre zéro implique la correspondance de Landau-Ginzburg/Calabi-Yau. Nous nous servons de ce résultat pour prouver la correspondance de Landau-Ginzburg/Calabi-Yau pour le type de Fermat, généralisant les résultats de A. Chiodo et Y. Ruan de [6].

We establish a new relationship (the MLK correspondence) between twisted FJRW theory and local Gromov-Witten theory in all genera. As a consequence, we show that the Landau-Ginzburg/Calabi-Yau correspondence is implied by the crepant transformation conjecture for Fermat type in genus zero. We use this to then prove the Landau-Ginzburg/Calabi-Yau correspondence for Fermat type, generalizing the results of A. Chiodo and Y. Ruan in [7].

Publié le :
DOI : 10.24033/asens.2312
Classification : 14N35, 14A20, 14E16, 53D45
Keywords: Crepant resolution, Landau-Ginzburg/Calabi-Yau correspondence, mirror symmetry, MLK correspondence.
Mot clés : Résolution crépante, correspondance de Landau-Ginzburg/Calabi-Yau, symétrie miroir, correspondance MLK. 
@article{ASENS_2016__49_6_1403_0,
     author = {Lee, Yuan-Pin and Priddis, Nathan and Shoemaker, Mark},
     title = {A proof of the {Landau-Ginzburg/} {Calabi-Yau} correspondence via  the crepant transformation conjecture},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1403--1443},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 49},
     number = {6},
     year = {2016},
     doi = {10.24033/asens.2312},
     mrnumber = {3592361},
     zbl = {1360.14133},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2312/}
}
TY  - JOUR
AU  - Lee, Yuan-Pin
AU  - Priddis, Nathan
AU  - Shoemaker, Mark
TI  - A proof of the Landau-Ginzburg/ Calabi-Yau correspondence via  the crepant transformation conjecture
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2016
SP  - 1403
EP  - 1443
VL  - 49
IS  - 6
PB  - Société Mathématique de France. Tous droits réservés
UR  - http://www.numdam.org/articles/10.24033/asens.2312/
DO  - 10.24033/asens.2312
LA  - en
ID  - ASENS_2016__49_6_1403_0
ER  - 
%0 Journal Article
%A Lee, Yuan-Pin
%A Priddis, Nathan
%A Shoemaker, Mark
%T A proof of the Landau-Ginzburg/ Calabi-Yau correspondence via  the crepant transformation conjecture
%J Annales scientifiques de l'École Normale Supérieure
%D 2016
%P 1403-1443
%V 49
%N 6
%I Société Mathématique de France. Tous droits réservés
%U http://www.numdam.org/articles/10.24033/asens.2312/
%R 10.24033/asens.2312
%G en
%F ASENS_2016__49_6_1403_0
Lee, Yuan-Pin; Priddis, Nathan; Shoemaker, Mark. A proof of the Landau-Ginzburg/ Calabi-Yau correspondence via  the crepant transformation conjecture. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 6, pp. 1403-1443. doi : 10.24033/asens.2312. http://www.numdam.org/articles/10.24033/asens.2312/

Abramovich, D.; Graber, T.; Vistoli, A. Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math., Volume 130 (2008), pp. 1337-1398 | DOI | MR | Zbl

Cadman, C. Using stacks to impose tangency conditions on curves, Amer. J. Math., Volume 129 (2007), pp. 405-427 (ISSN: 0002-9327) | DOI | MR | Zbl

Coates, T.; Corti, A.; Iritani, H.; Tseng, H.-H. Some Applications of the Mirror Theorem for Toric Stacks (preprint arXiv:1401.2611 ) | MR

Coates, T.; Corti, A.; Iritani, H.; Tseng, H.-H. Computing genus-zero twisted Gromov-Witten invariants, Duke Math. J., Volume 147 (2009), pp. 377-438 | DOI | MR | Zbl

Coates, T.; Corti, A.; Iritani, H.; Tseng, H.-H. A mirror theorem for toric stacks, Compos. Math., Volume 151 (2015), pp. 1878-1912 (ISSN: 0010-437X) | DOI | MR | Zbl

Coates, T.; Givental, A. Quantum Riemann-Roch, Lefschetz and Serre, Ann. of Math., Volume 165 (2007), pp. 15-53 (ISSN: 0003-486X) | DOI | MR | Zbl

Chiodo, A. Towards an enumerative geometry of the moduli space of twisted curves and rth roots, Compos. Math., Volume 144 (2008), pp. 1461-1496 (ISSN: 0010-437X) | DOI | MR | Zbl

Coates, T.; Iritani, H.; Jiang, Y. The Crepant Transformation Conjecture for Toric Complete Intersections (preprint arXiv:1410.0024 ) | MR

Chiodo, A.; Iritani, H.; Ruan, Y. Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence, Publ. Math. IHÉS, Volume 119 (2014), pp. 127-216 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Coates, T.; Iritani, H.; Tseng, H.-H. Wall-crossings in toric Gromov-Witten theory. I. Crepant examples, Geom. Topol., Volume 13 (2009), pp. 2675-2744 (ISSN: 1465-3060) | DOI | MR | Zbl

Clader, E. Landau-Ginzburg/Calabi-Yau correspondence for the complete intersections X 3 , 3 and X 2 , 2 , 2 , 2 (preprint arXiv:1301.5530 ) | MR

Coates, T. The Quantum Lefschetz Principle for Vector Bundles as a Map Between Givental Cones (preprint arXiv:1405.2893 )

Clader, E.; Priddis, N.; Shoemaker, M. Geometric Quantization with Applications to Gromov-Witten Theory (preprint arXiv:1309.1150 ) | MR

Chen, W.; Ruan, Y., Orbifolds in mathematics and physics (Madison, WI, 2001) (Contemp. Math.), Volume 310, Amer. Math. Soc., 2002, pp. 25-85 | DOI | MR | Zbl

Chen, W.; Ruan, Y. A new cohomology theory of orbifold, Comm. Math. Phys., Volume 248 (2004), pp. 1-31 | DOI | MR | Zbl

Chiodo, A.; Ruan, Y. Landau-Ginzburg/Calabi-Yau correspondence for quintic three-folds via symplectic transformations, Invent. math., Volume 182 (2010), pp. 117-165 | DOI | MR | Zbl

Coates, T.; Ruan, Y. Quantum cohomology and crepant resolutions: a conjecture, Ann. Inst. Fourier (Grenoble), Volume 63 (2013), pp. 431-478 http://aif.cedram.org/item?id=AIF_2013__63_2_431_0 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

Chiodo, A.; Zvonkine, D. Twisted r-spin potential and Givental's quantization, Adv. Theor. Math. Phys., Volume 13 (2009), pp. 1335-1369 http://projecteuclid.org/getRecord?id=euclid.atmp/1282054097 (ISSN: 1095-0761) | DOI | MR | Zbl

Fan, H.; Jarvis, T.; Ruan, Y. The Witten equation, mirror symmetry, and quantum singularity theory, Ann. of Math., Volume 178 (2013), pp. 1-106 | DOI | MR | Zbl

Givental, A. B. Semisimple Frobenius structures at higher genus, Int. Math. Res. Not., Volume 2001 (2001), pp. 1265-1286 (ISSN: 1073-7928) | DOI | MR | Zbl

Givental, A. B., Frobenius manifolds (Aspects Math., E36), Friedr. Vieweg, Wiesbaden, 2004, pp. 91-112 | DOI | MR | Zbl

Givental, A. B. Equivariant Gromov-Witten invariants, Int. Math. Res. Not., Volume 1996 (1996), pp. 613-663 | DOI | MR | Zbl

Hori, K.; Katz, S.; Klemm, A.; Pandharipande, R.; Thomas, R.; Vafa, C.; Vakil, R.; Zaslow, E., Clay Mathematics Monographs, 1, Amer. Math. Soc., Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2003, 929 pages (ISBN: 0-8218-2955-6) | MR | Zbl

Horja, R. P. Hypergeometric functions and mirror symmetry in toric varieties (preprint arXiv:math/9912109 ) | MR

Iritani, H. An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math., Volume 222 (2009), pp. 1016-1079 (ISSN: 0001-8708) | DOI | MR | Zbl

Iritani, H., New developments in algebraic geometry, integrable systems and mirror symmetry (RIMS, Kyoto, 2008) (Adv. Stud. Pure Math.), Volume 59, Math. Soc. Japan, Tokyo, 2010, pp. 111-166 | DOI | MR | Zbl

Lee, Y.-P., Algebraic geometry—Seattle 2005. Part 1 (Proc. Sympos. Pure Math.), Volume 80, Amer. Math. Soc., Providence, RI, 2009, pp. 309-323 | MR | Zbl

Lee, Y.-P.; Lin, H.-W.; Qu, F.; Wang, C.-L. Invariance of quantum rings under ordinary flops III: A quantum splitting principle, Camb. J. Math., Volume 4 (2016), pp. 333-401 | DOI | MR | Zbl

Priddis, N.; Shoemaker, M. A Landau-Ginzburg/Calabi-Yau correspondence for the mirror quintic, Ann. Inst. Fourier (Grenoble), Volume 66 (2016), pp. 1045-1091 http://aif.cedram.org/item?id=AIF_2016__66_3_1045_0 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

Polishchuk, A.; Vaintrob, A. Matrix factorizations and cohomological field theories, J. reine angew. Math., Volume 714 (2016), pp. 1-122 (ISSN: 0075-4102) | DOI | MR | Zbl

Teleman, C. The structure of 2D semi-simple field theories, Invent. math., Volume 188 (2012), pp. 525-588 (ISSN: 0020-9910) | DOI | MR | Zbl

Tseng, H.-H. Orbifold quantum Riemann-Roch, Lefschetz and Serre, Geom. Topol., Volume 14 (2010), pp. 1-81 | DOI | MR | Zbl

Witten, E., Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, pp. 235-269 | MR | Zbl

Witten, E., Mirror symmetry, II (AMS/IP Stud. Adv. Math.), Volume 1, Amer. Math. Soc., 1997, pp. 143-211 | MR | Zbl

Cité par Sources :