zbMATH — the first resource for mathematics

Tropical refined curve counting from higher genera and lambda classes. (English) Zbl 07015696
Summary: Block and Göttsche have defined a \(q\)-number refinement of counts of tropical curves in \(\mathbb{R}^2\). Under the change of variables \(q=e^{iu}\), we show that the result is a generating series of higher genus log Gromov-Witten invariants with insertion of a lambda class. This gives a geometric interpretation of the Block-Göttsche invariants and makes their deformation invariance manifest.

14T05 Tropical geometry (MSC2010)
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
Full Text: DOI
[1] Abramovich, D., Chen, Q., Gillam, W., Marcus, S.: The evaluation space of logarithmic stable maps. arXiv preprint arXiv:1012.5416 (2010)
[2] Abramovich, D.; Chen, Q., Stable logarithmic maps to Deligne-Faltings pairs II, Asian J. Math., 18, 465-488, (2014) · Zbl 1321.14025
[3] Abramovich, D., Chen, Q., Gross, M., Siebert, B.: Decomposition of degenerate Gromov-Witten invariants. arxiv preprint arXiv:1709.09864 (2017)
[4] Abramovich, D., Chen, Q., Gross, M., Siebert, B.: Punctured logarithmic curves. Preprint, available on the webpage of M. Gross (2017)
[5] Abramovich, D.; Marcus, S.; Wise, J., Comparison theorems for Gromov-Witten invariants of smooth pairs and of degenerations, Ann. Inst. Fourier (Grenoble), 64, 1611-1667, (2014) · Zbl 1317.14123
[6] Abramovich, D., Wise, J.: Birational invariance in logarithmic Gromov-Witten theory. arXiv preprint arXiv:1306.1222 (2013) · Zbl 1420.14124
[7] Behrend, K.; Fantechi, B., The intrinsic normal cone, Invent. Math., 128, 45-88, (1997) · Zbl 0909.14006
[8] Block, F.; Göttsche, L., Refined curve counting with tropical geometry, Compos. Math., 152, 115-151, (2016) · Zbl 1348.14125
[9] Bousseau, P.: The quantum tropical vertex. arXiv preprint arXiv:1806.11495 (2018)
[10] Bousseau, P.: Quantum mirrors of log Calabi-Yau surfaces and higher genus curves counting. arXiv preprint arXiv:1808.07336 (2018)
[11] Bryan, J., Oberdieck, G., Pandharipande, R., Yin, Q.: Curve counting on abelian surfaces and threefolds. arXiv preprint arXiv:1506.00841 (2015) · Zbl 1425.14044
[12] Bryan, J.; Pandharipande, R., Curves in Calabi-Yau threefolds and topological quantum field theory, Duke Math. J., 126, 369-396, (2005) · Zbl 1084.14053
[13] Chen, Q., The degeneration formula for logarithmic expanded degenerations, J. Algebr. Geom., 23, 341-392, (2014) · Zbl 1288.14039
[14] Chen, Q., Stable logarithmic maps to Deligne-Faltings pairs I, Ann. Math. (2), 180, 455-521, (2014) · Zbl 1311.14028
[15] Filippini, SA; Stoppa, J., Block-Göttsche invariants from wall-crossing, Compos. Math., 151, 1543-1567, (2015) · Zbl 1408.14179
[16] Fulton, W.: Intersection theory, volume 2 of 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), 2nd edn. Springer, Berlin (1998)
[17] Getzler, E.; Pandharipande, R., Virasoro constraints and the Chern classes of the Hodge bundle, Nuclear Phys. B, 530, 701-714, (1998) · Zbl 0957.14038
[18] Göttsche, L., Schroeter, F.: Refined broccoli invariants. arXiv preprint arXiv 1606, 09631 (2016)
[19] Göttsche, L.; Shende, V., Refined curve counting on complex surfaces, Geom. Topol., 18, 2245-2307, (2014) · Zbl 1310.14012
[20] Göttsche, L.; Shende, V., The \(\chi _{-y}\)-genera of relative Hilbert schemes for linear systems on Abelian and K3 surfaces, Algebr. Geom., 2, 405-421, (2015) · Zbl 1332.14009
[21] Gross, M.: Tropical geometry and mirror symmetry, volume 114 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (2011) · Zbl 1215.14061
[22] Gross, M.; Hacking, P.; Keel, S., Mirror symmetry for log Calabi-Yau surfaces I, Publ. Math. Inst. Hautes Études Sci., 122, 65-168, (2015) · Zbl 1351.14024
[23] Gross, M.; Pandharipande, R.; Siebert, B., The tropical vertex, Duke Math. J., 153, 297-362, (2010) · Zbl 1205.14069
[24] Gross, M.; Siebert, B., Logarithmic Gromov-Witten invariants, J. Am. Math. Soc., 26, 451-510, (2013) · Zbl 1281.14044
[25] Itenberg, I.; Mikhalkin, G., On Block-Göttsche multiplicities for planar tropical curves, Int. Math. Res. Not. IMRN, 23, 5289-5320, (2013) · Zbl 1329.14114
[26] Kato, K.: Logarithmic structures of Fontaine-Illusie. Algebraic analysis. geometry, and number theory (Baltimore, MD, 1988), pp. 191-224. Johns Hopkins Univ. Press, Baltimore (1989) · Zbl 0776.14004
[27] Kim, B., Lho, H., Ruddat, H.: The degeneration formula for stable log maps. arXiv preprint arXiv:1803.04210 (2018)
[28] Li, J., A degeneration formula of GW-invariants, J. Differ. Geom., 60, 199-293, (2002) · Zbl 1063.14069
[29] Manolache, C., Virtual pull-backs, J. Algebr. Geom., 21, 201-245, (2012) · Zbl 1328.14019
[30] Mandel, T., Ruddat, H.: Descendant log Gromov-Witten invariants for toric varieties and tropical curves. arXiv preprint arXiv:1612.02402 (2016)
[31] Maulik, D.; Nekrasov, N.; Okounkov, A.; Pandharipande, R., Gromov-Witten theory and Donaldson-Thomas theory. I, Compos. Math., 142, 1263-1285, (2006) · Zbl 1108.14046
[32] Maulik, D.; Nekrasov, N.; Okounkov, A.; Pandharipande, R., Gromov-Witten theory and Donaldson-Thomas theory. II., Compos. Math., 142, 1286-1304, (2006) · Zbl 1108.14047
[33] Maulik, D., Pandharipande, R., Thomas, R. P.: Curves on \(K3\) surfaces and modular forms. J. Topol. 3(4):937-996 (2010) (with an appendix by A. Pixton) · Zbl 1207.14058
[34] Mikhalkin, G., Enumerative tropical algebraic geometry in \(\mathbb{R}^2\), J. Am. Math. Soc., 18, 313-377, (2005) · Zbl 1092.14068
[35] Mikhalkin, G.: Quantum indices of real plane curves and refined enumerative geometry. arXiv preprint arXiv:1505.04338 (2015)
[36] Mumford, D.: Towards an enumerative geometry of the moduli space of curves. In: Arithmetic and Geometry, Vol. II, volume 36 of Progr. Math., pp. 271-328. Birkhäuser Boston, Boston, MA (1983) · Zbl 0554.14008
[37] Nicaise, J., Payne, S., Schroeter, F.: Tropical refined curve counting via motivic integration. arXiv preprint arXiv:1603.08424 (2016) · Zbl 1430.14037
[38] Nishinou, T.; Siebert, B., Toric degenerations of toric varieties and tropical curves, Duke Math. J., 135, 1-51, (2006) · Zbl 1105.14073
[39] Oberdieck, G., Pandharipande, R.: Curve counting on \(K3\times E\), the Igusa cusp form \(\chi _{10}\), and descendent integration. In: K3 surfaces and their moduli, volume 315 of Progr. Math., pp. 245-278. Birkhäuser/Springer, Cham (2016) · Zbl 1349.14176
[40] Olsson, M., Logarithmic geometry and algebraic stacks, Ann. Sci. École Norm. Sup. (4), 36, 747-791, (2003) · Zbl 1069.14022
[41] Pandharipande, R., Hodge integrals and degenerate contributions, Commun. Math. Phys., 208, 489-506, (1999) · Zbl 0953.14036
[42] Parker, B.: Three dimensional tropical correspondence formula. arXiv preprint arXiv:1608.02306 (2016) · Zbl 1401.14239
[43] Welschinger, J-Y, Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry, Invent. Math., 162, 195-234, (2005) · Zbl 1082.14052
[44] Zinger, A., A comparison theorem for Gromov-Witten invariants in the symplectic category, Adv. Math., 228, 535-574, (2011) · Zbl 1225.14046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.