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/

[1] Abramovich, Dan; Chen, Qile Stable logarithmic maps to Deligne–Faltings pairs II, Asian J. Math., Volume 18 (2014) no. 3, pp. 465-488 | DOI | MR | Zbl

[2] Abramovich, Dan; Chen, Qile; Gross, Mark; Siebert, Bernd Decomposition of degenerate Gromov-Witten invariants, Compos. Math., Volume 156 (2020) no. 10, pp. 2020-2075 | DOI | MR | Zbl

[3] Argüz, Hülya Real loci in (log-) Calabi–Yau manifolds via Kato–Nakayama spaces of toric degenerations (2016) | arXiv

[4] Argüz, Hülya Log geometric techniques for open invariants in mirror symmetry, Ph. D. Thesis, University of Hamburg (2017)

[5] Argüz, Hülya A tropical and log geometric approach to algebraic structures in the ring of theta functions (2017) | arXiv

[6] Argüz, Hülya Topological torus fibrations on Calabi–Yau manifolds via Kato–Nakayama spaces (2020) (to appear in Eur. J. Math.) | arXiv

[7] Bousseau, Pierrick A proof of N. Takahashi’s conjecture for (ℙ 2 ,E) and a refined sheaves/Gromov–Witten correspondence (2019) | arXiv

[8] Bousseau, Pierrick The quantum tropical vertex, Geom. Topol., Volume 24 (2020) no. 3, pp. 1297-1379 | DOI | MR | Zbl

[9] Brugallé, Erwan; Mikhalkin, Grigory Enumeration of curves via floor diagrams, C. R. Math. Acad. Sci. Paris, Volume 345 (2007) no. 6, pp. 329-334 | DOI | MR | Zbl

[10] Brugallé, Erwan; Mikhalkin, Grigory Floor decompositions of tropical curves: the planar case, Proceedings of Gökova Geometry-Topology Conference 2008 (2009), pp. 64-90 | MR | Zbl

[11] Cheung, Man-Wai; Fantini, Lorenzo; Park, Jennifer; Ulirsch, Martin Faithful realizability of tropical curves, Int. Math. Res. Not. (2016) no. 15, pp. 4706-4727 | DOI | MR | Zbl

[12] Filippini, Sara Angela; Stoppa, Jacopo Block–Göttsche invariants from wall-crossing, Compos. Math., Volume 151 (2015) no. 8, pp. 1543-1567 | DOI | MR | Zbl

[13] Fulton, William Introduction to toric varieties, Annals of Mathematics Studies, 131, Princeton University Press, 1993, xii+157 pages | DOI | MR

[14] Gathmann, Andreas; Markwig, Hannah; Schroeter, Franziska Broccoli curves and the tropical invariance of Welschinger numbers, Adv. Math., Volume 240 (2013), pp. 520-574 | DOI | MR | Zbl

[15] Georgieva, Penka; Zinger, Aleksey Real Gromov–Witten theory in all genera and real enumerative geometry: construction, Ann. Math., Volume 188 (2018) no. 3, pp. 685-752 | MR | Zbl

[16] Georgieva, Penka; Zinger, Aleksey Real Gromov–Witten theory in all genera and real enumerative geometry: computation, J. Differ. Geom., Volume 113 (2019) no. 3, pp. 417-491 | DOI | MR | Zbl

[17] Georgieva, Penka; Zinger, Aleksey Real Gromov–Witten theory in all genera and real enumerative geometry: properties, J. Symplectic Geom., Volume 17 (2019) no. 4, pp. 1083-1158 | DOI | MR | Zbl

[18] Gross, Andreas Correspondence theorems via tropicalizations of moduli spaces, Commun. Contemp. Math., Volume 18 (2016) no. 3, 1550043, 36 pages | DOI | MR | Zbl

[19] Gross, Mark Tropical geometry and mirror symmetry, CBMS Regional Conference Series in Mathematics, 114, American Mathematical Society, 2011, xvi+317 pages | DOI | MR

[20] Gross, Mark; Pandharipande, Rahul; Siebert, Bernd The tropical vertex, Duke Math. J., Volume 153 (2010) no. 2, pp. 297-362 | DOI | MR | Zbl

[21] Gross, Mark; Siebert, Bernd Mirror symmetry via logarithmic degeneration data. I, J. Differ. Geom., Volume 72 (2006) no. 2, pp. 169-338 | MR | Zbl

[22] Gross, Mark; Siebert, Bernd Logarithmic Gromov–Witten invariants, J. Am. Math. Soc., Volume 26 (2013) no. 2, pp. 451-510 | DOI | MR | Zbl

[23] Itenberg, Ilia; Kharlamov, Viatcheslav; Shustin, Eugenii Welschinger invariant and enumeration of real rational curves, Int. Math. Res. Not. (2003) no. 49, pp. 2639-2653 | DOI | MR | Zbl

[24] Itenberg, Ilia; Kharlamov, Viatcheslav; Shustin, Eugenii Logarithmic equivalence of the Welschinger and the Gromov–Witten invariants, Usp. Mat. Nauk, Volume 59 (2004) no. 6(360), pp. 85-110 | DOI | MR

[25] Itenberg, Ilia; Kharlamov, Viatcheslav; Shustin, Eugenii A Caporaso–Harris type formula for Welschinger invariants of real toric del Pezzo surfaces, Comment. Math. Helv., Volume 84 (2009) no. 1, pp. 87-126 | DOI | MR | Zbl

[26] Itenberg, Ilia; Kharlamov, Viatcheslav; Shustin, Eugenii Welschinger invariants revisited, Analysis meets geometry (Trends in Mathematics), Birkhäuser/Springer, 2017, pp. 239-260 | DOI | MR | Zbl

[27] Itenberg, Ilia; Mikhalkin, Grigory; Shustin, Eugenii Tropical algebraic geometry, Oberwolfach Seminars, 35, Birkhäuser, 2009, x+104 pages | DOI | MR

[28] Kato, Fumiharu Log smooth deformation theory, Tôhoku Math. J., Volume 48 (1996) no. 3, pp. 317-354 | DOI | MR | Zbl

[29] Kato, Fumiharu Log smooth deformation and moduli of log smooth curves, Int. J. Math., Volume 11 (2000) no. 2, pp. 215-232 | DOI | MR | Zbl

[30] Kato, Kazuya Logarithmic structures of Fontaine–Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins University Press, 1989, pp. 191-224 | MR | Zbl

[31] Mandel, Travis; Ruddat, Helge Descendant log Gromov–Witten invariants for toric varieties and tropical curves, Trans. Am. Math. Soc., Volume 373 (2020) no. 2, pp. 1109-1152 | DOI | MR | Zbl

[32] Mikhalkin, Grigory Enumerative tropical algebraic geometry in ℝ 2 , J. Am. Math. Soc., Volume 18 (2005) no. 2, pp. 313-377 | DOI | MR | Zbl

[33] Nishinou, Takeo Correspondence theorems for tropical curves (2009) | arXiv

[34] Nishinou, Takeo; Siebert, Bernd Toric degenerations of toric varieties and tropical curves, Duke Math. J., Volume 135 (2006) no. 1, pp. 1-51 | DOI | MR | Zbl

[35] Ogus, Arthur Lectures on logarithmic algebraic geometry, Cambridge Studies in Advanced Mathematics, 178, Cambridge University Press, 2018, xviii+539 pages | DOI | MR

[36] Ranganathan, Dhruv Skeletons of stable maps I: rational curves in toric varieties, J. Lond. Math. Soc., Volume 95 (2017) no. 3, pp. 804-832 | DOI | MR | Zbl

[37] Shustin, Eugenii A tropical approach to enumerative geometry, Algebra Anal., Volume 17 (2005) no. 2, pp. 170-214 | DOI | MR

[38] Shustin, Eugenii A tropical calculation of the Welschinger invariants of real toric del Pezzo surfaces, J. Algebr. Geom., Volume 15 (2006) no. 2, pp. 285-322 | DOI | MR | Zbl

[39] Shustin, Eugenii; Tyomkin, Ilya Patchworking singular algebraic curves. I, Isr. J. Math., Volume 151 (2006), pp. 125-144 | DOI | MR | Zbl

[40] Shustin, Eugenii; Tyomkin, Ilya Patchworking singular algebraic curves. II, Isr. J. Math., Volume 151 (2006), pp. 145-166 | DOI | MR | Zbl

[41] Solomon, Jake Philip Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions, Ph. D. Thesis, Massachusetts Institute of Technology (2006) (https://www.proquest.com/docview/304947035) | MR

[42] Solomon, Jake Philip; Tukachinsky, Sara Point like bounding chains in open Gromov–Witten theory (2016) | arXiv

[43] Tyomkin, Ilya Tropical geometry and correspondence theorems via toric stacks, Math. Ann., Volume 353 (2012) no. 3, pp. 945-995 | DOI | MR | Zbl

[44] Tyomkin, Ilya Enumeration of rational curves with cross-ratio constraints, Adv. Math., Volume 305 (2017), pp. 1356-1383 | DOI | MR | Zbl

[45] Viro, Oleg Yanovich Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7, Topology (Leningrad, 1982) (Lecture Notes in Mathematics), Volume 1060, Springer, 1984, pp. 187-200 | DOI | MR | Zbl

[46] Viro, Oleg Yanovich Real plane algebraic curves: constructions with controlled topology, Algebra Anal., Volume 1 (1989) no. 5, pp. 1-73 | MR

[47] Viro, Oleg Yanovich Patchworking real algebraic varieties (2006) | arXiv

[48] Weibel, Charles Alexander An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, 1994, xiv+450 pages | DOI | MR

[49] Welschinger, Jean-Yves Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry, C. R. Math. Acad. Sci. Paris, Volume 336 (2003) no. 4, pp. 341-344 | DOI | MR | Zbl

[50] Welschinger, Jean-Yves Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry, Invent. Math., Volume 162 (2005) no. 1, pp. 195-234 | DOI | MR | Zbl

[51] Welschinger, Jean-Yves Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants, Duke Math. J., Volume 127 (2005) no. 1, pp. 89-121 | DOI | MR | Zbl

[52] Welschinger, Jean-Yves Open Gromov–Witten invariants in dimension six, Math. Ann., Volume 356 (2013) no. 3, pp. 1163-1182 | DOI | MR | Zbl

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