Comparison theorems for Gromov–Witten invariants of smooth pairs and of degenerations

[Théorèmes de comparaison des invariants de Gromov-Witten de couples lisses et des dégénérescences]

Abramovich, Dan ; Marcus, Steffen ; Wise, Jonathan

Annales de l'Institut Fourier, Tome 64 (2014), p. 1611-1667 / Harvested from Numdam

Résumé

Nous considérons quatre approches à théorie de Gromov–Witten relative et à la théorie de Gromov-Witten des dégénérescences  : l’approche originale de J. Li, les expansions logarithmiques de B. Kim, les expansions orbifold de Abramovich–Fantechi, et une théorie logarithmique sans expansions de Gross–Siebert et Abramovich–Chen. Nous présentons quelques morphismes entre ces espaces et nous prouvons que leurs classes fondamentales virtuelles sont compatibles à travers ces morphismes. Par conséquent, les invariants de Gromov–Witten associés à chacune de ces quatre théories sont les mêmes.

Abstract

We consider four approaches to relative Gromov–Witten theory and Gromov–Witten theory of degenerations: J. Li’s original approach, B. Kim’s logarithmic expansions, Abramovich–Fantechi’s orbifold expansions, and a logarithmic theory without expansions due to Gross–Siebert and Abramovich–Chen. We exhibit morphisms relating these moduli spaces and prove that their virtual fundamental classes are compatible by pushforward through these morphisms. This implies that the Gromov–Witten invariants associated to all four of these theories are identical.

Détail

Publié le : 2014-01-01

MR 3329675 | Zbl 06387319

DOI : https://doi.org/10.5802/aif.2892
Classification:  14N35,  14H10,  14D23,  14D06,  14A20
Mots clés: géométrie algébrique, la théorie de Gromov–Witten, géométrie logarithmique, champs algébrique, espaces des modules, la théorie des deformations

Bibliographie

[1] Abramovich, D.; Cadman, C.; Fantechi, B.; Wise, J. Expanded degenerations and pairs, Communications in Algebra, Tome 41 (2013) no. 6, pp. 2346-2386 http://www.tandfonline.com/doi/abs/10.1080/00927872.2012.658589 | Article | MR 3225278

[2] Abramovich, D.; Cadman, C.; Wise, J. Relative and orbifold Gromov-Witten invariants (2010) (arXiv:1004.0981)

[3] Abramovich, D.; Chen, Q. Stable logarithmic maps to Deligne–Faltings pairs II, to appear (2013) (to appear. arXiv:1102.4531) | MR 3257836

[4] Abramovich, D.; Chen, Q.; Gillam, W. D.; Marcus, S. The Evaluation Space of Logarithmic Stable Maps (2010) (arXiv:1012.5416)

[5] Abramovich, D.; Corti, A.; Vistoli, A. Twisted bundles and admissible covers, Communications in Algebra, Tome 31 (2003) no. 8, pp. 3547-3618 http://www.tandfonline.com/doi/abs/10.1081/AGB-120022434 | Article | MR 2007376 | Zbl 1077.14034

[6] Abramovich, D.; Fantechi, B. Orbifold techniques in degeneration formulas (2011) (arXiv:1103.5132)

[7] Abramovich, Dan; Graber, Tom; Vistoli, Angelo Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math., Tome 130 (2008) no. 5, pp. 1337-1398 | Article | MR 2450211 | Zbl 1193.14070

[8] Abramovich, Dan; Vistoli, Angelo Compactifying the space of stable maps, J. Amer. Math. Soc., Tome 15 (2002) no. 1, pp. 27-75 | Article | MR 1862797 | Zbl 0991.14007

[9] Andreini, E.; Jiang, Y.; Tseng, H.-H. Gromov-Witten theory of banded gerbes over schemes (2011) (arXiv:1101.5996)

[10] Artin, M.; Grothendieck, A.; Verdier, J.-L. Théorie des topos et cohomologie étale des schémas, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 269, 270, 305 (1972)

[11] Behrend, K. The product formula for Gromov-Witten invariants, J. Algebraic Geom., Tome 8 (1999) no. 3, pp. 529-541 | MR 1689355 | Zbl 0938.14032

[12] Behrend, K.; Fantechi, B. The intrinsic normal cone, Invent. Math., Tome 128 (1997) no. 1, pp. 45-88 | Article | MR 1437495 | Zbl 0909.14006

[13] Breen, Lawrence Bitorseurs et cohomologie non abélienne, The Grothendieck Festschrift, Vol. I, Birkhäuser Boston, Boston, MA (Progr. Math.) Tome 86 (1990), pp. 401-476 | MR 1086889 | Zbl 0743.14034

[14] Cadman, Charles Using stacks to impose tangency conditions on curves, Amer. J. Math., Tome 129 (2007) no. 2, pp. 405-427 | Article | MR 2306040 | Zbl 1127.14002

[15] Cavalieri, Renzo; Marcus, Steffen; Wise, Jonathan Polynomial families of tautological classes on $\mathcal{M}_{g,n}^{rt}$, J. Pure Appl. Algebra, Tome 216 (2012) no. 4, pp. 950-981 | Article | MR 2864866 | Zbl 1273.14053

[16] Chen, Q. Stable logarithmic maps to Deligne-Faltings pairs I (2010) (arXiv:1008.3090) | MR 3224717

[17] Chen, W.; Ruan, Y. Orbifold Gromov-Witten theory, Orbifolds in mathematics and physics (Madison, WI, 2001), Tome J.53 (2005) no. 1, pp. 25-85 | MR 1950941 | Zbl 1091.53058

[18] Costello, Kevin Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products, Ann. of Math. (2), Tome 164 (2006) no. 2, pp. 561-601 | Article | MR 2247968 | Zbl 1209.14046

[19] Gillam, W. D. Logarithmic stacks and minimality (2011) (arXiv:1103.2140) | MR 2945649

[20] Gross, Mark; Siebert, Bernd Logarithmic Gromov-Witten invariants, J. Amer. Math. Soc., Tome 26 (2013) no. 2, pp. 451-510 | Article | MR 3011419 | Zbl 1281.14044

[21] Grothendieck, A. Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. (1961) no. 11, pp. 167 | Numdam | Zbl 0122.16102

[22] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. (1966) no. 28, pp. 255 | Numdam | Zbl 0135.39701

[23] Hall, Jack Openness of versality via coherent functors (2012) (arXiv.org:1206.4182)

[24] Ionel, Eleny-Nicoleta; Parker, Thomas H. Relative Gromov-Witten invariants, Ann. of Math. (2), Tome 157 (2003) no. 1, pp. 45-96 | Article | MR 1954264 | Zbl 1039.53101

[25] Ionel, Eleny-Nicoleta; Parker, Thomas H. The symplectic sum formula for Gromov-Witten invariants, Ann. of Math. (2), Tome 159 (2004) no. 3, pp. 935-1025 | Article | MR 2113018 | Zbl 1075.53092

[26] Kim, Bumsig Logarithmic stable maps, New developments in algebraic geometry, integrable systems and mirror symmetry (RIMS, Kyoto, 2008), Math. Soc. Japan, Tokyo (Adv. Stud. Pure Math.) Tome 59 (2010), pp. 167-200 | MR 2683209 | Zbl 1216.14023

[27] Li, An-Min; Ruan, Yongbin Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds, Invent. Math., Tome 145 (2001) no. 1, pp. 151-218 | Article | MR 1839289 | Zbl 1062.53073

[28] Li, Jun Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom., Tome 57 (2001) no. 3, pp. 509-578 | MR 1882667 | Zbl 1076.14540

[29] Li, Jun A degeneration formula of GW-invariants, J. Differential Geom., Tome 60 (2002) no. 2, pp. 199-293 | MR 1938113 | Zbl 1063.14069

[30] Manolache, Cristina Virtual pull-backs, J. Algebraic Geom., Tome 21 (2012) no. 2, pp. 201-245 | Article | MR 2877433

[31] Olsson, Martin C. Logarithmic geometry and algebraic stacks, Ann. Sci. École Norm. Sup. (4), Tome 36 (2003) no. 5, pp. 747-791 | Numdam | MR 2032986 | Zbl 1069.14022

[32] Olsson, Martin C. The logarithmic cotangent complex, Math. Ann., Tome 333 (2005) no. 4, pp. 859-931 | Article | MR 2195148 | Zbl 1095.14016

[33] Parker, B. Gromov Witten invariants of exploded manifolds (2011) (arXiv:1102.0158)

[34] Parker, Brett Exploded manifolds, Adv. Math., Tome 229 (2012) no. 6, pp. 3256-3319 | Article | MR 2900440 | Zbl 1276.53092

[35] Siebert, B. Gromov-Witten invariants in relative and singular cases (2001)

[36] Wise, J. Obstruction theories and virtual fundamental classes (2011) (arXiv:1111.4200)