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Descendant log Gromov-Witten invariants for toric varieties and tropical curves. (English) Zbl 1442.14166
The authors study genus zero, as well as higher genus Gromov-Witten invariants (in non-superabundant situations) of smooth toric varieties with Psi-class conditions. The main result shows that the tropical description of such invariants coincides with the classical one. The technique that is used to show this correspondence builds on the approach of T. Nishinou and B. Siebert [Duke Math. J. 135, No. 1, 1–51 (2006; Zbl 1105.14073)]. In particular, it uses logarithmic Gromov-Witten theory and toric degenerations. The authors also allow incidence conditions in the toric boundary for applications to non-toric situations. Moreover, tropically they study arbitrary tropical cycles as incidence conditions, not just affine linear ones as in Nishinou-Siebert.

##### MSC:
 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14N10 Enumerative problems (combinatorial problems) in algebraic geometry 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14T15 Combinatorial aspects of tropical varieties
##### Keywords:
tropical Gromov-Witten; Psi class insertion
##### Software:
ellipticcovers.lib
Full Text:
##### References:
 [1] Abramovich, Dan; Chen, Qile, Stable logarithmic maps to Deligne-Faltings pairs II, Asian J. Math., 18, 3, 465-488 (2014) · Zbl 1321.14025 [2] Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Pillip A., Geometry of algebraic curves. Volume II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 268, xxx+963 pp. (2011), Springer, Heidelberg · Zbl 1235.14002 [3] [ACGS17]ACGS D. Abramovich, Q. Chen, M. Gross, and B. Siebert, \emph Decomposition of degenerate gromov-witten invariants, arXiv:1709.09864 [math.AG], 2017. [4] Allermann, Lars; Rau, Johannes, First steps in tropical intersection theory, Math. Z., 264, 3, 633-670 (2010) · Zbl 1193.14074 [5] B\"ohm, Janko; Bringmann, Kathrin; Buchholz, Arne; Markwig, Hannah, Tropical mirror symmetry for elliptic curves, J. Reine Angew. Math., 732, 211-246 (2017) · Zbl 1390.14191 [6] Behrend, K.; Manin, Yu., Stacks of stable maps and Gromov-Witten invariants, Duke Math. J., 85, 1, 1-60 (1996) · Zbl 0872.14019 [7] Bousseau, Pierrick, Tropical refined curve counting from higher genera and lambda classes, Invent. Math., 215, 1, 1-79 (2019) · Zbl 07015696 [8] Cheung, Man-Wai; Fantini, Lorenzo; Park, Jennifer; Ulirsch, Martin, Faithful realizability of tropical curves, Int. Math. Res. Not. IMRN, 15, 4706-4727 (2016) · Zbl 1404.14071 [9] Cavalieri, Renzo; Johnson, Paul; Markwig, Hannah, Tropical Hurwitz numbers, J. Algebraic Combin., 32, 2, 241-265 (2010) · Zbl 1218.14058 [10] Cavalieri, Renzo; Johnson, Paul; Markwig, Hannah; Ranganathan, Dhruv, A graphical interface for the Gromov-Witten theory of curves. Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math. 97, 139-167 (2018), Amer. Math. Soc., Providence, RI [11] [CJMRb]CJMR2 Renzo Cavalieri, Paul Johnson, Hannah Markwig, and Dhruv Ranganathan, \emph Counting curves on toric surfaces: tropical geometry and the Fock space, arXiv:1706.05401, 2017. · Zbl 1391.14111 [12] Fantechi, B.; Pandharipande, R., Stable maps and branch divisors, Compositio Math., 130, 3, 345-364 (2002) · Zbl 1054.14033 [13] Fulton, William, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 2, xiv+470 pp. (1998), Springer-Verlag, Berlin · Zbl 0885.14002 [14] Gross, Mark; Hacking, Paul; Keel, Sean, Mirror symmetry for log Calabi-Yau surfaces I, Publ. Math. Inst. Hautes \'Etudes Sci., 122, 65-168 (2015) · Zbl 1351.14024 [15] Gross, Mark; Hacking, Paul; Keel, Sean; Kontsevich, Maxim, Canonical bases for cluster algebras, J. Amer. Math. Soc., 31, 2, 497-608 (2018) · Zbl 1446.13015 [16] Goulden, I. P.; Jackson, D. M.; Vakil, R., Towards the geometry of double Hurwitz numbers, Adv. Math., 198, 1, 43-92 (2005) · Zbl 1086.14022 [17] Gathmann, Andreas; Markwig, Hannah, The numbers of tropical plane curves through points in general position, J. Reine Angew. Math., 602, 155-177 (2007) · Zbl 1115.14049 [18] Gross, Mark; Pandharipande, Rahul; Siebert, Bernd, The tropical vertex, Duke Math. J., 153, 2, 297-362 (2010) · Zbl 1205.14069 [19] Gross, Mark, Mirror symmetry for $$\mathbb{P}^2$$ and tropical geometry, Adv. Math., 224, 1, 169-245 (2010) · Zbl 1190.14038 [20] Gross, Andreas, Intersection theory on tropicalizations of toroidal embeddings, Proc. Lond. Math. Soc. (3), 116, 6, 1365-1405 (2018) · Zbl 1420.14142 [21] Gross, Mark; Siebert, Bernd, Logarithmic Gromov-Witten invariants, J. Amer. Math. Soc., 26, 2, 451-510 (2013) · Zbl 1281.14044 [22] Hurwitz, A., Ueber die Anzahl der Riemann’schen Fl\"achen mit gegebenen Verzweigungspunkten, Math. Ann., 55, 1, 53-66 (1901) · JFM 32.0404.04 [23] [KLR]KLR B. Kim, H. Lho, and H. Ruddat, \emph The degeneration formula for stable log maps, https://arxiv.org/abs/1803.04210, 2018. [24] [Koc01]Kock J. Kock, \emph Notes on Psi classes, \urlhttp://mat.uab.cat/ kock/GW/notes/psi-notes.pdf, 2001. [25] [Mana]Man3 T. Mandel, \emph Refined tropical curve counts and canonical bases for quantum cluster algebras, arXiv:1503.06183, 2015. [26] [Manb]ManFrob T. Mandel, \emph Theta bases and log Gromov-Witten invariants of cluster varieties, arXiv:1903.03042, 2019. [27] Mikhalkin, Grigory, Enumerative tropical algebraic geometry in $$\mathbb{R}^2$$, J. Amer. Math. Soc., 18, 2, 313-377 (2005) · Zbl 1092.14068 [28] Mikhalkin, Grigory, Moduli spaces of rational tropical curves. Proceedings of G\"okova Geometry-Topology Conference 2006, 39-51 (2007), G\"okova Geometry/Topology Conference (GGT), G\"okova · Zbl 1203.14027 [29] [MRa]MakRud C. Y. Mak and H. Ruddat, \emph Tropically constructed Lagrangians in mirror quintic threefolds, arXiv:1904.11780, 2019. [30] [MRb]MRudMult T. Mandel and H. Ruddat, \emph Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves, arXiv:1902.07183, 2019. [31] Markwig, Hannah; Rau, Johannes, Tropical descendant Gromov-Witten invariants, Manuscripta Math., 129, 3, 293-335 (2009) · Zbl 1171.14039 [32] [Nis10]Ni T. Nishinou, \emph Correspondence theorems for tropical curves, arXiv:0912.5090, 2010. [33] Nishinou, Takeo; Siebert, Bernd, Toric degenerations of toric varieties and tropical curves, Duke Math. J., 135, 1, 1-51 (2006) · Zbl 1105.14073 [34] Overholser, Peter, A descendent tropical Landau-Ginzburg potential for $$\Bbb P^2$$, Commun. Number Theory Phys., 10, 4, 739-803 (2016) · Zbl 1415.14021 [35] Pandharipande, R., The Toda equations and the Gromov-Witten theory of the Riemann sphere, Lett. Math. Phys., 53, 1, 59-74 (2000) · Zbl 0999.14020 [36] Ranganathan, Dhruv, Skeletons of stable maps I: rational curves in toric varieties, J. Lond. Math. Soc. (2), 95, 3, 804-832 (2017) · Zbl 1401.14130 [37] Rau, Johannes, Intersections on tropical moduli spaces, Rocky Mountain J. Math., 46, 2, 581-662 (2016) · Zbl 1379.14035 [38] Vistoli, Angelo, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math., 97, 3, 613-670 (1989) · Zbl 0694.14001
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