Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

Request Permissions   Purchase Content 
 
 

 

Logarithmic Gromov-Witten invariants


Authors: Mark Gross and Bernd Siebert
Journal: J. Amer. Math. Soc. 26 (2013), 451-510
MSC (2010): Primary 14D20, 14N35
DOI: https://doi.org/10.1090/S0894-0347-2012-00757-7
Published electronically: November 20, 2012
MathSciNet review: 3011419
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The goal of this paper is to give a general theory of logarithmic Gromov-Witten invariants. This gives a vast generalization of the theory of relative Gromov-Witten invariants introduced by Li-Ruan, Ionel-Parker, and Jun Li and completes a program first proposed by the second named author in 2002. One considers target spaces $ X$ carrying a log structure. Domains of stable log curves are log smooth curves. Algebraicity of the stack of such stable log maps is proven, subject only to the hypothesis that the log structure on $ X$ is fine, saturated, and Zariski. A notion of basic stable log map is introduced; all stable log maps are pull-backs of basic stable log maps via base-change. With certain additional hypotheses, the stack of basic stable log maps is proven to be proper. Under these hypotheses and the additional hypothesis that $ X$ is log smooth, one obtains a theory of log Gromov-Witten invariants.


References [Enhancements On Off] (What's this?)

  • [AbCh] D. Abramovich, Q. Chen: Stable logarithmic maps to Deligne-Faltings pairs II, preprint arXiv:1102.4531 [math.AG], 19pp.
  • [ACGM] D. Abramovich, Q. Chen, W.D. Gillam, S. Marcus: The evaluation space of logarithmic stable maps, preprint arXiv:1012.5416 [math.AG], 19pp.
  • [AMW] D. Abramovich, S. Marcus, J. Wise: Comparison theorems for Gromov-Witten invariants of smooth pairs and of degenerations, preprint arXiv:1207.2085 [math.AG], 43pp.
  • [Ar] M. Artin, Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165-189. MR 0399094 (53 #2945)
  • [Be] K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127 (1997), no. 3, 601-617. MR 1431140 (98i:14015), https://doi.org/10.1007/s002220050132
  • [BeFa] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45-88. MR 1437495 (98e:14022), https://doi.org/10.1007/s002220050136
  • [BeMa] K. Behrend and Yu. Manin, Stacks of stable maps and Gromov-Witten invariants, Duke Math. J. 85 (1996), no. 1, 1-60. MR 1412436 (98i:14014), https://doi.org/10.1215/S0012-7094-96-08501-4
  • [Ch] Q. Chen: Stable logarithmic maps to Deligne-Faltings pairs I, preprint arXiv:1008.3090 [math.AG], 48pp.
  • [DeMu] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75-109. MR 0262240 (41 #6850)
  • [FuPa] W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, Algebraic geometry--Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 45-96. MR 1492534 (98m:14025)
  • [Ga] Andreas Gathmann, Absolute and relative Gromov-Witten invariants of very ample hypersurfaces, Duke Math. J. 115 (2002), no. 2, 171-203. MR 1944571 (2003k:14068), https://doi.org/10.1215/S0012-7094-02-11521-X
  • [Gs] Mark Gross, The Strominger-Yau-Zaslow conjecture: from torus fibrations to degenerations, Algebraic geometry--Seattle 2005. Part 1, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence, RI, 2009, pp. 149-192. MR 2483935 (2010b:14080)
  • [Gt] Alexander Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert, Séminaire Bourbaki, Vol. 6, Soc. Math. France, Paris, 1995, pp. Exp. No. 221, 249-276 (French). MR 1611822
  • [Il] Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin, 1971 (French). MR 0491680 (58 #10886a)
  • [IoPa] 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), https://doi.org/10.4007/annals.2003.157.45
  • [Kf] Fumiharu Kato, Log smooth deformation and moduli of log smooth curves, Internat. J. Math. 11 (2000), no. 2, 215-232. MR 1754621 (2001d:14016), https://doi.org/10.1142/S0129167X0000012X
  • [Kk] 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)
  • [Ki] 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)
  • [Kd] Finn F. Knudsen, The projectivity of the moduli space of stable curves. II. The stacks $ M_{g,n}$, Math. Scand. 52 (1983), no. 2, 161-199. MR 702953 (85d:14038a)
  • [Kt] Donald Knutson, Algebraic spaces, Lecture Notes in Mathematics, Vol. 203, Springer-Verlag, Berlin, 1971. MR 0302647 (46 #1791)
  • [Kr] Andrew Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), no. 3, 495-536. MR 1719823 (2001a:14003), https://doi.org/10.1007/s002220050351
  • [LaMB] 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)
  • [LiRu] 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), https://doi.org/10.1007/s002220100146
  • [Li1] Jun Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001), no. 3, 509-578. MR 1882667 (2003d:14066)
  • [Li2] Jun Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), no. 2, 199-293. MR 1938113 (2004k:14096)
  • [LiTi] Jun Li and Gang Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), no. 1, 119-174. MR 1467172 (99d:14011), https://doi.org/10.1090/S0894-0347-98-00250-1
  • [NiSi] Takeo Nishinou and Bernd Siebert, Toric degenerations of toric varieties and tropical curves, Duke Math. J. 135 (2006), no. 1, 1-51. MR 2259922 (2007h:14083), https://doi.org/10.1215/S0012-7094-06-13511-1
  • [Nt] Nitin Nitsure, Construction of Hilbert and Quot schemes, Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 105-137. MR 2223407
  • [Og] A. Ogus: Lectures on logarithmic algebraic geometry. TeXed notes (2006).
  • [Ol1] Martin Christian Olsson, Log algebraic stacks and moduli of log schemes, ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)-University of California, Berkeley. MR 2702292
  • [Ol2] 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), https://doi.org/10.1016/j.ansens.2002.11.001
  • [Ol3] Martin C. Olsson, The logarithmic cotangent complex, Math. Ann. 333 (2005), no. 4, 859-931. MR 2195148 (2006j:14017), https://doi.org/10.1007/s00208-005-0707-6
  • [Ol4] Martin C. Olsson, Deformation theory of representable morphisms of algebraic stacks, Math. Z. 253 (2006), no. 1, 25-62. MR 2206635 (2006i:14010), https://doi.org/10.1007/s00209-005-0875-9
  • [Pa] Brett Parker, Exploded manifolds, Adv. Math. 229 (2012), no. 6, 3256-3319. MR 2900440, https://doi.org/10.1016/j.aim.2012.02.005
  • [SGA1] Revêtements étales et groupe fondamental, Springer-Verlag, Berlin, 1971 (French). Séminaire de Géométrie Algébrique du Bois Marie 1960-1961 (SGA 1); Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud; Lecture Notes in Mathematics, Vol. 224. MR 0354651 (50 #7129)
  • [Si1] Bernd Siebert, Virtual fundamental classes, global normal cones and Fulton's canonical classes, Frobenius manifolds, Aspects Math., E36, Vieweg, Wiesbaden, 2004, pp. 341-358. MR 2115776 (2005k:14119)
  • [Si2] B. Siebert: Gromov-Witten invariants in relative and singular cases, Lecture given at the workshop ``Algebraic aspects of mirror symmetry'', Univ. Kaiserslautern, Germany, June 2001.
  • [Si3] B. Siebert: Obstruction theories revisited, manuscript 2002.

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 14D20, 14N35

Retrieve articles in all journals with MSC (2010): 14D20, 14N35


Additional Information

Mark Gross
Affiliation: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0112
Email: mgross@math.ucsd.edu

Bernd Siebert
Affiliation: FB Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany
Email: bernd.siebert@math.uni-hamburg.de

DOI: https://doi.org/10.1090/S0894-0347-2012-00757-7
Received by editor(s): March 16, 2011
Received by editor(s) in revised form: August 26, 2011, and July 30, 2012
Published electronically: November 20, 2012
Additional Notes: This work was partially supported by NSF grants 0505325 and 0805328.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society