no. 4
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, Dan1 ; Marcus, Steffen2 ; Wise, Jonathan3
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.

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.

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

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. 
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.2892/

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