Abramovich, Dan; Marcus, Steffen; Wise, Jonathan
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 ]
Annales de l'institut Fourier, Tome 64 (2014) no. 4 , p. 1611-1667
MR 3329675 | Zbl 06387319
doi : 10.5802/aif.2892
URL stable : http://www.numdam.org/item?id=AIF_2014__64_4_1611_0

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
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.

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