# zbMATH — the first resource for mathematics

Decomposition of degenerate Gromov-Witten invariants. (English) Zbl 07283066
Summary: We prove a decomposition formula of logarithmic Gromov-Witten invariants in a degeneration setting. A one-parameter log smooth family $$X\longrightarrow B$$ with singular fibre over $$b_0\in B$$ yields a family $$\mathscr{M}(X/B,\beta)\longrightarrow B$$ of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over $$b_0$$ in terms of rigid tropical maps to the tropicalization of $$X/B$$. This generalizes one aspect of known results in the case that the fibre $$X_{b_0}$$ is a normal crossings union of two divisors. We exhibit our formulas in explicit examples.
##### MSC:
 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14D23 Stacks and moduli problems
Full Text:
##### References:
 [1] Abramovich, D., Caporaso, L. and Payne, S., The tropicalization of the moduli space of curves, Ann. Sci. Éc. Norm. Supér. (4)48 (2015), 765-809. · Zbl 1410.14049 [2] Abramovich, D. and Chen, Q., Stable logarithmic maps to Deligne-Faltings pairs II, Asian J. Math. 18 (2014), 465-488. · Zbl 1321.14025 [3] Abramovich, D., Chen, Q., Gross, M. and Siebert, B., Punctured logarithmic maps, Preprint (2020), arXiv:2009.07720. [4] Abramovich, D., Chen, Q., Marcus, S. and Wise, J., Boundedness of the space of stable logarithmic maps, J. Eur. Math. Soc. (JEMS)19 (2017), 2783-2809. · Zbl 1453.14081 [5] Abramovich, D. and Karu, K., Weak semistable reduction in characteristic 0, Invent. Math. 139 (2000), 241-273. · Zbl 0958.14006 [6] Abramovich, D. and Wise, J., Birational invariance in logarithmic Gromov-Witten theory, Compos. Math. 154 (2018), 595-620. · Zbl 1420.14124 [7] Arbarello, E., Cornalba, M. and Griffiths, P., Geometry of algebraic curves, Volume II (Springer, 2011). · Zbl 1235.14002 [8] Behrend, K., Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), 601-617. · Zbl 0909.14007 [9] Behrend, K. and Fantechi, B., The intrinsic normal cone, Invent. Math. 128 (1997), 45-88. · Zbl 0909.14006 [10] Behrend, K. and Manin, Y., Stacks of stable maps and Gromov-Witten invariants, Duke Math. J. 85 (1996), 1-60. · Zbl 0872.14019 [11] Bryan, J. and Leung, N. C., The enumerative geometry of $$K3$$ surfaces and modular forms, J. Amer. Math. Soc. 13 (2000), 371-410. · Zbl 0963.14031 [12] Burgos Gil, J. and Sombra, M., When do the recession cones of a polyhedral complex form a fan?Discrete Comput. Geom. 46 (2011), 789-798. · Zbl 1233.14031 [13] Cadman, C., Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), 405-427. · Zbl 1127.14002 [14] Cavalieri, R., Chan, M., Ulirsch, M. and Wise, J., A moduli stack of tropical curves, Forum Math. Sigma 8 (2020), e23. · Zbl 1444.14005 [15] Chen, Q., Stable logarithmic maps to Deligne-Faltings pairs I, Ann. of Math. (2)180 (2014), 455-521. · Zbl 1311.14028 [16] Costello, K., Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products, Ann. of Math. (2)164 (2006), 561-601. · Zbl 1209.14046 [17] Gross, M. and Siebert, B., Logarithmic Gromov-Witten invariants, J. Amer. Math. Soc. 26 (2013), 451-510. · Zbl 1281.14044 [18] Gross, M. and Siebert, B., Intrinsic mirror symmetry, Preprint (2019), arXiv:1909.07649. · Zbl 1448.14039 [19] Kato, K., Logarithmic structures of Fontaine-Illusie, in Algebraic analysis, geometry, and number theory, Baltimore, MD, 1988 (Johns Hopkins University Press, 1989), 191-224. · Zbl 0776.14004 [20] Kato, F., Exactness, integrality, and log modifications, Preprint (1999), arXiv:math/9907124. [21] Kato, F., Log smooth deformation and moduli of log smooth curves, Internat. J. Math. 11 (2000), 215-232. · Zbl 1100.14502 [22] Kempf, G., Knudsen, F., Mumford, D. and Saint-Donat, B., Toroidal embeddings I, , vol. 339 (Springer, 1973). · Zbl 0271.14017 [23] Kim, B., Lho, H. and Ruddat, H., The degeneration formula for stable log maps, Preprint (2018), arXiv:1803.04210. [24] Knudsen, F., The projectivity of the moduli space of stable curves. II. The stacks $$M_{g,n}$$, Math. Scand. 52 (1983), 161-199. · Zbl 0544.14020 [25] Kresch, A., Cycle groups for Artin stacks, Invent. Math. 138 (1999), 495-536. · Zbl 0938.14003 [26] Li, J., A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), 199-293. · Zbl 1063.14069 [27] Mandel, T. and Ruddat, H., Descendant log Gromov-Witten invariants for toric varieties and tropical curves, Trans. Amer. Math. Soc.373 (2020), 1109-1152. · Zbl 1442.14166 [28] Manolache, C., Virtual pull-backs, J. Algebraic Geom. 21 (2012), 201-245. · Zbl 1328.14019 [29] Mikhalkin, G., Enumerative tropical algebraic geometry in $$\mathbb{R}^2$$, J. Amer. Math. Soc. 18 (2005), 313-377. · Zbl 1092.14068 [30] Mochizuki, S., The geometry of the compactification of the Hurwitz scheme, Publ. Res. Inst. Math. Sci. 31 (1995), 355-441. · Zbl 0866.14013 [31] Nishinou, T. and Siebert, B., Toric degenerations of toric varieties and tropical curves, Duke Math. J. 135 (2006), 1-51. · Zbl 1105.14073 [32] Ogus, A., Lectures on logarithmic algebraic geometry (Cambridge University Press, 2018). · Zbl 1437.14003 [33] Olsson, M., Logarithmic geometry and algebraic stacks, Ann. Sci. École Norm. Sup. (4)36 (2003), 747-791. · Zbl 1069.14022 [34] Parker, B., Holomorphic curves in exploded manifolds: compactness, Adv. Math.283 (2015), 377-457. · Zbl 1322.32013 [35] Parker, B., Tropical gluing formulae for Gromov-Witten invariants, Preprint (2017), arXiv:1703.05433. [36] Parker, B., Holomorphic curves in exploded manifolds: regularity, Geom. Topol. 23 (2019), 1621-1690. · Zbl 1421.32042 [37] Parker, B., Holomorphic curves in exploded manifolds: Kuranishi structure, Preprint (2019), arXiv:1301.4748. · Zbl 1421.32042 [38] Parker, B., Holomorphic curves in exploded manifolds: virtual fundamental class, Geom. Topol. 23 (2019), 1877-1960. · Zbl 1421.53086 [39] Ranganathan, D., Logarithmic Gromov-Witten theory with expansions, Preprint (2020), arXiv:1903.09006. [40] The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu (2017). [41] Ulirsch, M., Tropical geometry of logarithmic schemes, PhD thesis, Brown University (2015). · Zbl 1349.14197 [42] Ulirsch, M., Functorial tropicalization of logarithmic schemes: the case of constant coefficients, Proc. Lond. Math. Soc. (3) 114 (2017), 1081-1113. · Zbl 1419.14088 [43] Ulirsch, M., A non-Archimedean analogue of Teichmüller space and its tropicalization, Preprint (2020), arXiv:2004.07508. [44] Wise, J., Moduli of morphisms of logarithmic schemes, Algebra Number Theory10 (2016), 695-735. · Zbl 1343.14020 [45] Wise, J., Uniqueness of minimal morphisms of logarithmic schemes, Algebr. Geom. 6 (2019), 50-63. · Zbl 1441.14007 [46] Yu, T. Y., Enumeration of holomorphic cylinders in log Calabi-Yau surfaces. II. Positivity, integrality and the gluing formula, Geom. Topol., to appear. Preprint (2020), arXiv:1608.07651.
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.