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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

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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

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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

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