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.
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.
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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
@article{AIF_2022__72_4_1547_0, author = {Arg\"uz, H\"ulya and Bousseau, Pierrick}, 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}, volume = {72}, number = {4}, year = {2022}, doi = {10.5802/aif.3507}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3507/} }
TY - JOUR AU - Argüz, Hülya AU - Bousseau, Pierrick TI - Real Log Curves in Toric Varieties, Tropical Curves, and Log Welschinger Invariants JO - Annales de l'Institut Fourier PY - 2022 SP - 1547 EP - 1620 VL - 72 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3507/ DO - 10.5802/aif.3507 LA - en ID - AIF_2022__72_4_1547_0 ER -
%0 Journal Article %A Argüz, Hülya %A Bousseau, Pierrick %T Real Log Curves in Toric Varieties, Tropical Curves, and Log Welschinger Invariants %J Annales de l'Institut Fourier %D 2022 %P 1547-1620 %V 72 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3507/ %R 10.5802/aif.3507 %G en %F AIF_2022__72_4_1547_0
Argüz, Hülya; Bousseau, Pierrick. Real Log Curves in Toric Varieties, Tropical Curves, and Log Welschinger Invariants. Annales de l'Institut Fourier, Volume 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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