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

Dan Abramovich[1]; Steffen Marcus[2]; Jonathan Wise[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 (2014)

  • Volume: 64, Issue: 4, page 1611-1667
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Abramovich, Dan, Marcus, Steffen, and Wise, Jonathan. "Comparison theorems for Gromov–Witten invariants of smooth pairs and of degenerations." Annales de l’institut Fourier 64.4 (2014): 1611-1667. <http://eudml.org/doc/275513>.

@article{Abramovich2014,
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.},
affiliation = {Department of Mathematics Brown University Box 1917 Providence, RI 02912 U.S.A.; Department of Mathematics University of Utah 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 U.S.A.; Department of Mathematics University of Colorado at Boulder Campus Box 395 Boulder, CO 80309-0395 U.S.A.},
author = {Abramovich, Dan, Marcus, Steffen, Wise, Jonathan},
journal = {Annales de l’institut Fourier},
keywords = {algberaic geometry; Gromov–Witten theory; logarithmic geometry; algebraic stacks; moduli spaces; deformation theory; Gromov-Witten theory},
language = {eng},
number = {4},
pages = {1611-1667},
publisher = {Association des Annales de l’institut Fourier},
title = {Comparison theorems for Gromov–Witten invariants of smooth pairs and of degenerations},
url = {http://eudml.org/doc/275513},
volume = {64},
year = {2014},
}

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
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 4
SP - 1611
EP - 1667
AB - 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.
LA - eng
KW - algberaic geometry; Gromov–Witten theory; logarithmic geometry; algebraic stacks; moduli spaces; deformation theory; Gromov-Witten theory
UR - http://eudml.org/doc/275513
ER -

References

top
  1. D. Abramovich, C. Cadman, B. Fantechi, J. Wise, Expanded degenerations and pairs, Communications in Algebra 41 (2013), 2346-2386 Zbl1326.14020MR3225278
  2. D. Abramovich, C. Cadman, J. Wise, Relative and orbifold Gromov-Witten invariants, (2010) 
  3. D. Abramovich, Q. Chen, Stable logarithmic maps to Deligne–Faltings pairs II, to appear (2013) Zbl1321.14025MR3257836
  4. D. Abramovich, Q. Chen, W. D. Gillam, S. Marcus, The Evaluation Space of Logarithmic Stable Maps, (2010) 
  5. D. Abramovich, A. Corti, A. Vistoli, Twisted bundles and admissible covers, Communications in Algebra 31 (2003), 3547-3618 Zbl1077.14034MR2007376
  6. D. Abramovich, B. Fantechi, Orbifold techniques in degeneration formulas, (2011) 
  7. Dan Abramovich, Tom Graber, Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), 1337-1398 Zbl1193.14070MR2450211
  8. Dan Abramovich, Angelo Vistoli, Compactifying the space of stable maps, J. Amer. Math. Soc. 15 (2002), 27-75 Zbl0991.14007MR1862797
  9. E. Andreini, Y. Jiang, H.-H. Tseng, Gromov-Witten theory of banded gerbes over schemes, (2011) 
  10. M. Artin, A. Grothendieck, J.-L. Verdier, Théorie des topos et cohomologie étale des schémas, 269, 270, 305 (1972), Springer-Verlag, Berlin Zbl0234.00007
  11. K. Behrend, The product formula for Gromov-Witten invariants, J. Algebraic Geom. 8 (1999), 529-541 Zbl0938.14032MR1689355
  12. K. Behrend, B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), 45-88 Zbl0909.14006MR1437495
  13. Lawrence Breen, Bitorseurs et cohomologie non abélienne, The Grothendieck Festschrift, Vol. I 86 (1990), 401-476, Birkhäuser Boston, Boston, MA Zbl0743.14034MR1086889
  14. Charles Cadman, Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), 405-427 Zbl1127.14002MR2306040
  15. Renzo Cavalieri, Steffen Marcus, Jonathan Wise, Polynomial families of tautological classes on ℳ g , n r t , J. Pure Appl. Algebra 216 (2012), 950-981 Zbl1273.14053MR2864866
  16. Q. Chen, Stable logarithmic maps to Deligne-Faltings pairs I, (2010) Zbl1311.14028MR3224717
  17. W. Chen, Y. Ruan, Orbifold Gromov-Witten theory, Orbifolds in mathematics and physics (Madison, WI, 2001) J.53 (2005), 25-85 Zbl1091.53058MR1950941
  18. Kevin Costello, Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products, Ann. of Math. (2) 164 (2006), 561-601 Zbl1209.14046MR2247968
  19. W. D. Gillam, Logarithmic stacks and minimality, (2011) Zbl1248.18008MR2945649
  20. Mark Gross, Bernd Siebert, Logarithmic Gromov-Witten invariants, J. Amer. Math. Soc. 26 (2013), 451-510 Zbl1281.14044MR3011419
  21. A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. (1961) Zbl0122.16102
  22. A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. (1966) Zbl0135.39701
  23. Jack Hall, Openness of versality via coherent functors, (2012) 
  24. Eleny-Nicoleta Ionel, Thomas H. Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003), 45-96 Zbl1039.53101MR1954264
  25. Eleny-Nicoleta Ionel, Thomas H. Parker, The symplectic sum formula for Gromov-Witten invariants, Ann. of Math. (2) 159 (2004), 935-1025 Zbl1075.53092MR2113018
  26. Bumsig Kim, Logarithmic stable maps, New developments in algebraic geometry, integrable systems and mirror symmetry (RIMS, Kyoto, 2008) 59 (2010), 167-200, Math. Soc. Japan, Tokyo Zbl1216.14023MR2683209
  27. An-Min Li, Yongbin Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds, Invent. Math. 145 (2001), 151-218 Zbl1062.53073MR1839289
  28. Jun Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001), 509-578 Zbl1076.14540MR1882667
  29. Jun Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), 199-293 Zbl1063.14069MR1938113
  30. Cristina Manolache, Virtual pull-backs, J. Algebraic Geom. 21 (2012), 201-245 Zbl1328.14019MR2877433
  31. Martin C. Olsson, Logarithmic geometry and algebraic stacks, Ann. Sci. École Norm. Sup. (4) 36 (2003), 747-791 Zbl1069.14022MR2032986
  32. Martin C. Olsson, The logarithmic cotangent complex, Math. Ann. 333 (2005), 859-931 Zbl1095.14016MR2195148
  33. B. Parker, Gromov Witten invariants of exploded manifolds, (2011) 
  34. Brett Parker, Exploded manifolds, Adv. Math. 229 (2012), 3256-3319 Zbl1276.53092MR2900440
  35. B. Siebert, Gromov-Witten invariants in relative and singular cases, (2001) 
  36. J. Wise, Obstruction theories and virtual fundamental classes, (2011) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.