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The degeneration formula for logarithmic expanded degenerations. (English) Zbl 1288.14039
A degeneration formula for Gromov-Witten invariants was first proven by Jun Li in algebraic settings. He developed the stack of expanded degenerations and the notion of predeformable maps to prove the degeneration formula. One difficulty in Jun Li’s approach is that predeformability is not an open condition in general. Jun Li studied the deformation theory of predeformable maps using log structures for the construction of the virtual fundamental class without using the technology of logarithmic cotangent complex which was developed later on by Olsson. Recently, Abramovich and Fantechi introduced the notion of transversal maps using the theory of root stacks. Transversality is an open condition and hence the construction of the perfect obstruction theory and virtual fundamental class is more natural in this approach. Abramovich and Fantechi then proved a degeneration formula using the transversal maps.
The paper under review proves a degeneration formula for Gromov-Witten invariants inspired by Kim’s theory of log stable maps. Interestingly, the degeneration formula of this paper is very similar to that of Abramovich and Fantechi and the author of the paper under review expects that the theory of transversal maps and the theory of log stable maps are equivalent.

MSC:
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14D06 Fibrations, degenerations in algebraic geometry
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References:
[1] Dan Abramovich and Qile Chen, Stable logarithmic maps to Deligne-Faltings pairs II, arXiv:1102.4531 (2011); to appear, Asian J. Math. · Zbl 1321.14025
[2] Dan Abramovich, Charles Cadman, Barbara Fantechi, and Jonathan Wise, On the moduli stacks of expanded degenerations and pairs, arXiv:1110.2976v1 (2011). · Zbl 1326.14020
[3] Dan Abramovich, Charles Cadman, and Jonathan Wise, Relative and orbifold Gromov-Witten invariants, arXiv:1004.0981v1 (2010). · Zbl 06849616
[4] Dan Abramovich and Barbara Fantechi, Orbifold techniques in degeneration formulas, arXiv:1103.5132v1 (2011). · Zbl 1375.14182
[5] Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337 – 1398. · Zbl 1193.14070
[6] M. Artin, Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165 – 189. · Zbl 0317.14001
[7] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45 – 88. · Zbl 0909.14006
[8] Jim Bryan and Naichung Conan Leung, The enumerative geometry of \?3 surfaces and modular forms, J. Amer. Math. Soc. 13 (2000), no. 2, 371 – 410. · Zbl 0963.14031
[9] A. Bondal and D. Orlov, Semiorthogonal decompositions for algebraic varieties, arXiv:alb-beom/9506012 (1995).
[10] Niels Borne and Angelo Vistoli, Parabolic sheaves on logarithmic schemes, Adv. Math. 231 (2012), no. 3-4, 1327 – 1363. · Zbl 1256.14002
[11] Charles Cadman, Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), no. 2, 405 – 427. · Zbl 1127.14002
[12] Qile Chen, Stable logarithmic maps to Deligne-Faltings pairs I, arXiv:1008.3090 (2010); to appear, Annals of Math. · Zbl 1311.14028
[13] David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, vol. 68, American Mathematical Society, Providence, RI, 1999. · Zbl 0951.14026
[14] Kevin Costello, Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products, Ann. of Math. (2) 164 (2006), no. 2, 561 – 601. · Zbl 1209.14046
[15] Weimin Chen and Yongbin Ruan, Orbifold Gromov-Witten theory, Orbifolds in mathematics and physics (Madison, WI, 2001) Contemp. Math., vol. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 25 – 85. · Zbl 1091.53058
[16] William Fulton, Intersection theory, 2nd ed., 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], vol. 2, Springer-Verlag, Berlin, 1998. · Zbl 0885.14002
[17] Mark Gross and Bernd Siebert, Logarithmic Gromov-Witten invariants,J. Amer. Math. Soc. 26 (2013), no. 2, 451-510. · Zbl 1281.14044
[18] Eleny-Nicoleta Ionel, GW Invariants Relative Normal Crossings Divisors, arXiv:1103.3977 (2011). · Zbl 1349.57006
[19] Eleny-Nicoleta Ionel and Thomas H. Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003), no. 1, 45 – 96. · Zbl 1039.53101
[20] Eleny-Nicoleta Ionel and Thomas H. Parker, The symplectic sum formula for Gromov-Witten invariants, Ann. of Math. (2) 159 (2004), no. 3, 935 – 1025. · Zbl 1075.53092
[21] Kazuya Kato, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 191 – 224. · Zbl 0776.14004
[22] Fumiharu Kato, Log smooth deformation and moduli of log smooth curves, Internat. J. Math. 11 (2000), no. 2, 215 – 232. · Zbl 1100.14502
[23] Bumsig Kim, Logarithmic stable maps, New developments in algebraic geometry, integrable systems and mirror symmetry (RIMS, Kyoto, 2008) Adv. Stud. Pure Math., vol. 59, Math. Soc. Japan, Tokyo, 2010, pp. 167 – 200. · Zbl 1216.14023
[24] Andrew Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), no. 3, 495 – 536. · Zbl 0938.14003
[25] Jun Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001), no. 3, 509 – 578. · Zbl 1076.14540
[26] Jun Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199 – 293. · Zbl 1063.14069
[27] Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, 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], vol. 39, Springer-Verlag, Berlin, 2000 (French). · Zbl 0945.14005
[28] An-Min Li and Yongbin Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds, Invent. Math. 145 (2001), no. 1, 151 – 218. · Zbl 1062.53073
[29] Kenji Matsuki and Martin Olsson, Kawamata-Viehweg vanishing as Kodaira vanishing for stacks, Math. Res. Lett. 12 (2005), no. 2-3, 207 – 217. · Zbl 1080.14023
[30] Shinichi Mochizuki, The geometry of the compactification of the Hurwitz scheme, Publ. Res. Inst. Math. Sci. 31 (1995), no. 3, 355 – 441. · Zbl 0866.14013
[31] Arthur Ogus, Lectures on logarithmic algebraic geometry, TeXed notes (2001). · Zbl 1028.14013
[32] Martin C. Olsson, Logarithmic geometry and algebraic stacks, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 5, 747 – 791 (English, with English and French summaries). · Zbl 1069.14022
[33] Martin C. Olsson, Universal log structures on semi-stable varieties, Tohoku Math. J. (2) 55 (2003), no. 3, 397 – 438. · Zbl 1069.14015
[34] Martin C. Olsson, The logarithmic cotangent complex, Math. Ann. 333 (2005), no. 4, 859 – 931. · Zbl 1095.14016
[35] Brett Parker, Exploded manifolds, Adv. Math. 229 (2012), no. 6, 3256 – 3319. · Zbl 1276.53092
[36] Brett Parker, Holomorphic curves in exploded manifolds: Compactness, arXiv:0911.2241v1 (2009). · Zbl 1322.32013
[37] Brett Parker, Holomorphic curves in exploded torus fibrations: Regularity, arXiv:0902.0087v1 (2009).
[38] Bernd Siebert, Gromov-Witten invariants in relative and singular cases, Lecture given in the workshop on algebraic aspects of mirror symmetry, Universität Kaiserslautern, Germany, June 26, 2001.
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