no. 4
Real Log Curves in Toric Varieties, Tropical Curves, and Log Welschinger Invariants
[Log-courbes réelles dans les variétés toriques, courbes tropicales, et log-invariants de Welschinger]
Argüz, Hülya1 ; Bousseau, Pierrick2
1 Laboratoire de Mathématiques Université de Versailles St Quentin en Yvelines (France)
2 Université Paris-Saclay, CNRS Laboratoire de mathématiques d’Orsay 91405 Orsay (France)
Annales de l'Institut Fourier, Tome 72 (2022) no. 4, pp. 1547-1620.

Nous donnons une description tropicale du comptage des log-ourbes réelles dans les dégénérescences toriques de variétés toriques. Nous traitons le cas des courbes de genre zéro et toutes les situations non-superabondantes pour les courbes de genre supérieur. La preuve repose sur la théorie des déformations logarithmiques et est une version réelle de l’approche par Nishinou–Siebert du théorème de correspondance tropicale pour les courbes complexes. En dimension deux, nous utilisons des techniques similaires pour étudier le comptage de log-courbes réelles avec signes de Welschinger et nous obtenons une nouvelle preuve du théorème de correspondance tropicale de Mikhalkin pour les invariants de Weslchinger.

We give a tropical description of the counting of real log curves in toric degenerations of toric varieties. We treat the case of genus zero curves and all non-superabundant higher-genus situations. The proof relies on log deformation theory and is a real version of the Nishinou–Siebert approach to the tropical correspondence theorem for complex curves. In dimension two, we use similar techniques to study the counting of real log curves with Welschinger signs and we obtain a new proof of Mikhalkin’s tropical correspondence theorem for Welschinger invariants.

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DOI : 10.5802/aif.3507
Classification : 14T05, 14N10, 14N35, 14P99
Keywords: log Gromov–Witten invariants, Welschinger invariants, toric varieties, tropical geometry, real geometry
Mot clés : Invariants log-Gromov–Witten, invariants de Welschinger, variétés toriques, géométrie tropicale, géométrie réelle

Argüz, Hülya 1 ; Bousseau, Pierrick 2

1 Laboratoire de Mathématiques Université de Versailles St Quentin en Yvelines (France)
2 Université Paris-Saclay, CNRS Laboratoire de mathématiques d’Orsay 91405 Orsay (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Real {Log} {Curves} in {Toric} {Varieties,} {Tropical} {Curves,} and {Log} {Welschinger} {Invariants}},
     journal = {Annales de l'Institut Fourier},
     pages = {1547--1620},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Argüz, Hülya; Bousseau, Pierrick. Real Log Curves in Toric Varieties, Tropical Curves, and Log Welschinger Invariants. Annales de l'Institut Fourier, Tome 72 (2022) no. 4, pp. 1547-1620. doi : 10.5802/aif.3507. https://aif.centre-mersenne.org/articles/10.5802/aif.3507/

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