The degeneration formula for logarithmic expanded degenerations
Author:
Qile Chen
Journal:
J. Algebraic Geom. 23 (2014), 341-392
DOI:
https://doi.org/10.1090/S1056-3911-2013-00614-1
Published electronically:
November 15, 2013
MathSciNet review:
3166394
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We define a notion of logarithmic stable maps based on Bumsig Kim’s construction. Then we prove Jun Li’s degeneration formula under the log setting by applying the method for transversal maps developed by Dan Abramovich and Barbara Fantechi.
References
- Dan Abramovich and Qile Chen, Stable logarithmic maps to Deligne–Faltings pairs II, arXiv:1102.4531 (2011); to appear, Asian J. Math. *15pt
- Dan Abramovich, Charles Cadman, Barbara Fantechi, and Jonathan Wise, On the moduli stacks of expanded degenerations and pairs, arXiv:1110.2976v1 (2011).
- Dan Abramovich, Charles Cadman, and Jonathan Wise, Relative and orbifold Gromov-Witten invariants, arXiv:1004.0981v1 (2010).
- Dan Abramovich and Barbara Fantechi, Orbifold techniques in degeneration formulas, arXiv:1103.5132v1 (2011).
- Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398. MR 2450211, DOI https://doi.org/10.1353/ajm.0.0017
- M. Artin, Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165–189. MR 399094, DOI https://doi.org/10.1007/BF01390174
- K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88. MR 1437495, DOI https://doi.org/10.1007/s002220050136
- Jim Bryan and Naichung Conan Leung, The enumerative geometry of $K3$ surfaces and modular forms, J. Amer. Math. Soc. 13 (2000), no. 2, 371–410. MR 1750955, DOI https://doi.org/10.1090/S0894-0347-00-00326-X
- A. Bondal and D. Orlov, Semiorthogonal decompositions for algebraic varieties, arXiv:alb-beom/9506012 (1995).
- Niels Borne and Angelo Vistoli, Parabolic sheaves on logarithmic schemes, Adv. Math. 231 (2012), no. 3-4, 1327–1363. MR 2964607, DOI https://doi.org/10.1016/j.aim.2012.06.015
- Charles Cadman, Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), no. 2, 405–427. MR 2306040, DOI https://doi.org/10.1353/ajm.2007.0007
- Qile Chen, Stable logarithmic maps to Deligne–Faltings pairs I, arXiv:1008.3090 (2010); to appear, Annals of Math.
- David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, vol. 68, American Mathematical Society, Providence, RI, 1999. MR 1677117
- 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 https://doi.org/10.4007/annals.2006.164.561
- 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. MR 1950941, DOI https://doi.org/10.1090/conm/310/05398
- 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. MR 1644323
- Mark Gross and Bernd Siebert, Logarithmic Gromov-Witten invariants, J. Amer. Math. Soc. 26 (2013), no. 2, 451–510.
- Eleny-Nicoleta Ionel, GW Invariants Relative Normal Crossings Divisors, arXiv:1103.3977 (2011). *1\baselineskip
- Eleny-Nicoleta Ionel and Thomas H. Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003), no. 1, 45–96. MR 1954264, DOI https://doi.org/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 https://doi.org/10.4007/annals.2004.159.935
- 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. MR 1463703
- Fumiharu Kato, Log smooth deformation and moduli of log smooth curves, Internat. J. Math. 11 (2000), no. 2, 215–232. MR 1754621, DOI https://doi.org/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 https://doi.org/10.2969/aspm/05910167
- Andrew Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), no. 3, 495–536. MR 1719823, DOI https://doi.org/10.1007/s002220050351
- 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
- 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). MR 1771927
- 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 https://doi.org/10.1007/s002220100146
- Kenji Matsuki and Martin Olsson, Kawamata-Viehweg vanishing as Kodaira vanishing for stacks, Math. Res. Lett. 12 (2005), no. 2-3, 207–217. MR 2150877, DOI https://doi.org/10.4310/MRL.2005.v12.n2.a6
- Shinichi Mochizuki, The geometry of the compactification of the Hurwitz scheme, Publ. Res. Inst. Math. Sci. 31 (1995), no. 3, 355–441. MR 1355945, DOI https://doi.org/10.2977/prims/1195164048
- Arthur Ogus, Lectures on logarithmic algebraic geometry, TeXed notes (2001).
- 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). MR 2032986, DOI https://doi.org/10.1016/j.ansens.2002.11.001
- 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, The logarithmic cotangent complex, Math. Ann. 333 (2005), no. 4, 859–931. MR 2195148, DOI https://doi.org/10.1007/s00208-005-0707-6
- Brett Parker, Exploded manifolds, Adv. Math. 229 (2012), no. 6, 3256–3319. MR 2900440, DOI https://doi.org/10.1016/j.aim.2012.02.005
- Brett Parker, Holomorphic curves in exploded manifolds: Compactness, arXiv:0911.2241v1 (2009).
- Brett Parker, Holomorphic curves in exploded torus fibrations: Regularity, arXiv:0902.0087v1 (2009).
- 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.
References
- Dan Abramovich and Qile Chen, Stable logarithmic maps to Deligne–Faltings pairs II, arXiv:1102.4531 (2011); to appear, Asian J. Math. *15pt
- Dan Abramovich, Charles Cadman, Barbara Fantechi, and Jonathan Wise, On the moduli stacks of expanded degenerations and pairs, arXiv:1110.2976v1 (2011).
- Dan Abramovich, Charles Cadman, and Jonathan Wise, Relative and orbifold Gromov-Witten invariants, arXiv:1004.0981v1 (2010).
- Dan Abramovich and Barbara Fantechi, Orbifold techniques in degeneration formulas, arXiv:1103.5132v1 (2011).
- Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398. MR 2450211 (2009k:14108), DOI https://doi.org/10.1353/ajm.0.0017
- M. Artin, Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165–189. MR 0399094 (53 \#2945)
- K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88. MR 1437495 (98e:14022), DOI https://doi.org/10.1007/s002220050136
- Jim Bryan and Naichung Conan Leung, The enumerative geometry of $K3$ surfaces and modular forms, J. Amer. Math. Soc. 13 (2000), no. 2, 371–410 (electronic). MR 1750955 (2001i:14071), DOI https://doi.org/10.1090/S0894-0347-00-00326-X
- A. Bondal and D. Orlov, Semiorthogonal decompositions for algebraic varieties, arXiv:alb-beom/9506012 (1995).
- Niels Borne and Angelo Vistoli, Parabolic sheaves on logarithmic schemes, Adv. Math. 231 (2012), no. 3-4, 1327–1363. MR 2964607, DOI https://doi.org/10.1016/j.aim.2012.06.015
- Charles Cadman, Using stacks to impose tangency conditions on curves, Amer. J. Math. 129 (2007), no. 2, 405–427. MR 2306040 (2008g:14016), DOI https://doi.org/10.1353/ajm.2007.0007
- Qile Chen, Stable logarithmic maps to Deligne–Faltings pairs I, arXiv:1008.3090 (2010); to appear, Annals of Math.
- David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, vol. 68, American Mathematical Society, Providence, RI, 1999. MR 1677117 (2000d:14048)
- 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 (2007i:14057), DOI https://doi.org/10.4007/annals.2006.164.561
- 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. MR 1950941 (2004k:53145), DOI https://doi.org/10.1090/conm/310/05398
- 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. MR 1644323 (99d:14003)
- Mark Gross and Bernd Siebert, Logarithmic Gromov-Witten invariants, J. Amer. Math. Soc. 26 (2013), no. 2, 451–510.
- Eleny-Nicoleta Ionel, GW Invariants Relative Normal Crossings Divisors, arXiv:1103.3977 (2011). *1\baselineskip
- Eleny-Nicoleta Ionel and Thomas H. Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003), no. 1, 45–96. MR 1954264 (2004a:53112), DOI https://doi.org/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 (2006b:53110), DOI https://doi.org/10.4007/annals.2004.159.935
- 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. MR 1463703 (99b:14020)
- Fumiharu Kato, Log smooth deformation and moduli of log smooth curves, Internat. J. Math. 11 (2000), no. 2, 215–232. MR 1754621 (2001d:14016), DOI https://doi.org/10.1142/S0129167X0000012X
- Bumsig Kim, Logarithmic stable maps, Symmetry (RIMS, Kyoto, 2008) Adv. Stud. Pure Math., vol. 59, Math. Soc. Japan, Tokyo, 2010, pp. 167–200. MR 2683209 (2011m:14019)
- Andrew Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), no. 3, 495–536. MR 1719823 (2001a:14003), DOI https://doi.org/10.1007/s002220050351
- Jun Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001), no. 3, 509–578. MR 1882667 (2003d:14066)
- Jun Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199–293. MR 1938113 (2004k:14096)
- 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). MR 1771927 (2001f:14006)
- 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 (2002g:53158), DOI https://doi.org/10.1007/s002220100146
- Kenji Matsuki and Martin Olsson, Kawamata-Viehweg vanishing as Kodaira vanishing for stacks, Math. Res. Lett. 12 (2005), no. 2-3, 207–217. MR 2150877 (2006c:14023)
- Shinichi Mochizuki, The geometry of the compactification of the Hurwitz scheme, Publ. Res. Inst. Math. Sci. 31 (1995), no. 3, 355–441. MR 1355945 (96j:14017), DOI https://doi.org/10.2977/prims/1195164048
- Arthur Ogus, Lectures on logarithmic algebraic geometry, TeXed notes (2001).
- 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). MR 2032986 (2004k:14018), DOI https://doi.org/10.1016/j.ansens.2002.11.001
- Martin C. Olsson, Universal log structures on semi-stable varieties, Tohoku Math. J. (2) 55 (2003), no. 3, 397–438. MR 1993863 (2004f:14025)
- Martin C. Olsson, The logarithmic cotangent complex, Math. Ann. 333 (2005), no. 4, 859–931. MR 2195148 (2006j:14017), DOI https://doi.org/10.1007/s00208-005-0707-6
- Brett Parker, Exploded manifolds, Adv. Math. 229 (2012), no. 6, 3256–3319. MR 2900440, DOI https://doi.org/10.1016/j.aim.2012.02.005
*15pt
- Brett Parker, Holomorphic curves in exploded manifolds: Compactness, arXiv:0911.2241v1 (2009).
- Brett Parker, Holomorphic curves in exploded torus fibrations: Regularity, arXiv:0902.0087v1 (2009).
- 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.
Additional Information
Qile Chen
Affiliation:
Department of Mathematics, Box 1917, Brown University, Providence, Rhode Island 02912
Address at time of publication:
Department of Mathematics, Columbia University, Rm 628, MC 4421, 2990 Broadway, New York, New York 10027
MR Author ID:
924581
Email:
q.chen@math.brown.edu, q_chen@math.columbia.edu
Received by editor(s):
March 28, 2011
Received by editor(s) in revised form:
February 6, 2012
Published electronically:
November 15, 2013
Additional Notes:
Research partially supported by funds from NSF award DMS-0901278.
Article copyright:
© Copyright 2013
University Press, Inc.
The copyright for this article reverts to public domain 28 years after publication.