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

Abramovich, Dan; Marcus, Steffen; Wise, Jonathan

doi:10.5802/aif.2892

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 ¹ ; Marcus, Steffen ² ; Wise, Jonathan ³

¹ Department of Mathematics Brown University Box 1917 Providence, RI 02912 U.S.A.
² Department of Mathematics University of Utah 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 U.S.A.
³ Department of Mathematics University of Colorado at Boulder Campus Box 395 Boulder, CO 80309-0395 U.S.A.

Annales de l'Institut Fourier, Tome 64 (2014) no. 4, pp. 1611-1667.

Résumé
Abstract

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.

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.

MR Zbl

DOI : 10.5802/aif.2892

Classification : 14N35, 14H10, 14D23, 14D06, 14A20
Keywords: algberaic geometry, Gromov–Witten theory, logarithmic geometry, algebraic stacks, moduli spaces, deformation theory
Mot clés : géométrie algébrique, la théorie de Gromov–Witten, géométrie logarithmique, champs algébrique, espaces des modules, la théorie des deformations

Affiliations des auteurs :

Abramovich, Dan ¹ ; Marcus, Steffen ² ; Wise, Jonathan ³

@article{AIF_2014__64_4_1611_0,
     author = {Abramovich, Dan and Marcus, Steffen and Wise, Jonathan},
     title = {Comparison theorems for {Gromov{\textendash}Witten} invariants of smooth pairs and~of~degenerations},
     journal = {Annales de l'Institut Fourier},
     pages = {1611--1667},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {64},
     number = {4},
     year = {2014},
     doi = {10.5802/aif.2892},
     zbl = {06387319},
     mrnumber = {3329675},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2892/}
}

TY  - JOUR
AU  - Abramovich, Dan
AU  - Marcus, Steffen
AU  - Wise, Jonathan
TI  - Comparison theorems for Gromov–Witten invariants of smooth pairs and of degenerations
JO  - Annales de l'Institut Fourier
PY  - 2014
SP  - 1611
EP  - 1667
VL  - 64
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2892/
DO  - 10.5802/aif.2892
LA  - en
ID  - AIF_2014__64_4_1611_0
ER  -

%0 Journal Article
%A Abramovich, Dan
%A Marcus, Steffen
%A Wise, Jonathan
%T Comparison theorems for Gromov–Witten invariants of smooth pairs and of degenerations
%J Annales de l'Institut Fourier
%D 2014
%P 1611-1667
%V 64
%N 4
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2892/
%R 10.5802/aif.2892
%G en
%F AIF_2014__64_4_1611_0

Abramovich, Dan; Marcus, Steffen; Wise, Jonathan. Comparison theorems for Gromov–Witten invariants of smooth pairs and of degenerations. Annales de l'Institut Fourier, Tome 64 (2014) no. 4, pp. 1611-1667. doi : 10.5802/aif.2892. http://www.numdam.org/articles/10.5802/aif.2892/

Bibliographie
Cité par

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

[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

[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, Volume 31 (2003) no. 8, pp. 3547-3618 http://www.tandfonline.com/doi/abs/10.1081/AGB-120022434 | DOI | MR | Zbl

[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., Volume 130 (2008) no. 5, pp. 1337-1398 | DOI | MR | Zbl

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

[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, Lecture Notes in Mathematics, 269, 270, 305, Springer-Verlag, Berlin, 1972

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

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

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

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

[15] Cavalieri, Renzo; Marcus, Steffen; Wise, Jonathan Polynomial families of tautological classes on $ℳ_{g, n}^{r t}$ , J. Pure Appl. Algebra, Volume 216 (2012) no. 4, pp. 950-981 | DOI | MR | Zbl

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

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

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

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

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

[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

[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

[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), Volume 157 (2003) no. 1, pp. 45-96 | DOI | MR | Zbl

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

[26] Kim, Bumsig Logarithmic stable maps, 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. 167-200 | MR | Zbl

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

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

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

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

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

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

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

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

[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)

Cité par Sources :