Localization for logarithmic stable maps
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Abstract:
We prove a virtual localization formula for Bumsig Kim’s space of Logarithmic Stable Maps. The formula is closely related and can in fact recover the relative virtual localization formula of Graber-Vakil.References
- Dan Abramovich and Qile Chen, Stable logarithmic maps to Deligne-Faltings pairs II, Asian J. Math. 18 (2014), no. 3, 465–488. MR 3257836, DOI 10.4310/AJM.2014.v18.n3.a5
- Dan Abramovich, Steffen Marcus, and Jonathan Wise, Comparison theorems for Gromov-Witten invariants of smooth pairs and of degenerations, Ann. Inst. Fourier (Grenoble) 64 (2014), no. 4, 1611–1667 (English, with English and French summaries). MR 3329675, DOI 10.5802/aif.2892
- Dan Abramovich and Angelo Vistoli, Compactifying the space of stable maps, J. Amer. Math. Soc. 15 (2002), no. 1, 27–75. MR 1862797, DOI 10.1090/S0894-0347-01-00380-0
- Niels Borne and Angelo Vistoli, Parabolic sheaves on logarithmic schemes, Adv. Math. 231 (2012), no. 3-4, 1327–1363. MR 2964607, DOI 10.1016/j.aim.2012.06.015
- Qile Chen, Stable logarithmic maps to Deligne-Faltings pairs I, Ann. of Math. (2) 180 (2014), no. 2, 455–521. MR 3224717, DOI 10.4007/annals.2014.180.2.2
- Huai-Liang Chang, Young-Hoon Kiem, and Jun Li, Torus localization and wall crossing for cosection localized virtual cycles, Adv. Math. 308 (2017), 964–986. MR 3600080, DOI 10.1016/j.aim.2016.12.019
- Kevin Costello, Higher genus Gromov-Witten invariants as genus zero invariants of symmetric products, Ann. of Math. (2) 164 (2006), no. 2, 561–601. MR 2247968, DOI 10.4007/annals.2006.164.561
- W. D. Gillam, Logarithmic stacks and minimality, Internat. J. Math. 23 (2012), no. 7, 1250069, 38. MR 2945649, DOI 10.1142/S0129167X12500693
- T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518. MR 1666787, DOI 10.1007/s002220050293
- Mark Gross, Tropical geometry and mirror symmetry, CBMS Regional Conference Series in Mathematics, vol. 114, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2011. MR 2722115, DOI 10.1090/cbms/114
- Mark Gross and Bernd Siebert, Logarithmic Gromov-Witten invariants, J. Amer. Math. Soc. 26 (2013), no. 2, 451–510. MR 3011419, DOI 10.1090/S0894-0347-2012-00757-7
- Tom Graber and Ravi Vakil, Relative virtual localization and vanishing of tautological classes on moduli spaces of curves, Duke Math. J. 130 (2005), no. 1, 1–37. MR 2176546, DOI 10.1215/S0012-7094-05-13011-3
- Eleny-Nicoleta Ionel and Thomas H. Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003), no. 1, 45–96. MR 1954264, DOI 10.4007/annals.2003.157.45
- 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. MR 2113018, DOI 10.4007/annals.2004.159.935
- K. Kato, Logarithmic degeneration and Dieudonne theory, preprint.
- Fumiharu Kato, Log smooth deformation theory, Tohoku Math. J. (2) 48 (1996), no. 3, 317–354. MR 1404507, DOI 10.2748/tmj/1178225336
- Fumiharu Kato, Log smooth deformation and moduli of log smooth curves, Internat. J. Math. 11 (2000), no. 2, 215–232. MR 1754621, DOI 10.1142/S0129167X0000012X
- 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. MR 2683209, DOI 10.2969/aspm/05910167
- Maxim Kontsevich, Enumeration of rational curves via torus actions, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 335–368. MR 1363062, DOI 10.1007/978-1-4612-4264-2_{1}2
- Jun Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001), no. 3, 509–578. MR 1882667
- Jun Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199–293. MR 1938113
- 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. MR 1839289, DOI 10.1007/s002220100146
- Cristina Manolache, Virtual pull-backs, J. Algebraic Geom. 21 (2012), no. 2, 201–245. MR 2877433, DOI 10.1090/S1056-3911-2011-00606-1
- D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906, DOI 10.1007/978-3-642-57916-5
- Wiesława Nizioł, Toric singularities: log-blow-ups and global resolutions, J. Algebraic Geom. 15 (2006), no. 1, 1–29. MR 2177194, DOI 10.1090/S1056-3911-05-00409-1
- Martin C. Olsson, Universal log structures on semi-stable varieties, Tohoku Math. J. (2) 55 (2003), no. 3, 397–438. MR 1993863
- Martin C. Olsson, (Log) twisted curves, Compos. Math. 143 (2007), no. 2, 476–494. MR 2309994, DOI 10.1112/S0010437X06002442
- Matthieu Romagny, Group actions on stacks and applications, Michigan Math. J. 53 (2005), no. 1, 209–236. MR 2125542, DOI 10.1307/mmj/1114021093
- Burt Totaro, The resolution property for schemes and stacks, J. Reine Angew. Math. 577 (2004), 1–22. MR 2108211, DOI 10.1515/crll.2004.2004.577.1
- Angelo Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 (1989), no. 3, 613–670. MR 1005008, DOI 10.1007/BF01388892
Additional Information
- S. Molcho
- Affiliation: Department of Mathematics, University of Colorado Boulder, Campus Box 395, Boulder, Colorado 80309-0395; and Mathematical Research Center, Scuola Normale Superiore di Pisa 56126 Pisa, Italy
- MR Author ID: 1162839
- Email: samouil.molcho@sns.it
- E. Routis
- Affiliation: Department of Mathematics, Brown University, Box 1917, 151 Thayer Street, Providence, Rhode Island 02912; and Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
- Address at time of publication: Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
- MR Author ID: 1227537
- Email: evangelos.routis@ipmu.jp, routis@mpim-bonn.mpg.de
- Received by editor(s): November 19, 2015
- Received by editor(s) in revised form: April 10, 2018, and July 16, 2018
- Published electronically: February 4, 2019
- Additional Notes: The second author was supported by the World Premier International Research Center Initiative (WPI), MEXT, Japan
- © Copyright 2019 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 6 (2019), 80-113
- MSC (2010): Primary 14N35; Secondary 14D23
- DOI: https://doi.org/10.1090/btran/31
- MathSciNet review: 3905962