Splitting of Gromov–Witten invariants with toric gluing strata
Author:
Yixian Wu
Journal:
J. Algebraic Geom. 33 (2024), 213-263
DOI:
https://doi.org/10.1090/jag/826
Published electronically:
November 13, 2023
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We prove a splitting formula that reconstructs the logarithmic Gromov–Witten invariants of simple normal crossing varieties from the punctured Gromov–Witten invariants of their irreducible components, under the assumption of the gluing strata being toric varieties. The formula is based on the punctured Gromov–Witten theory developed by Abramovich, Chen, Gross, and Siebert.
References
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- Dan Abramovich and Qile Chen, Stable logarithmic maps to Deligne-Faltings pairs, II, Asian J. Math. 18 (2014), no. 3, 465–488., DOI 10.4310/AJM.2014.v18.n3.a5
- Dan Abramovich, Qile Chen, Mark Gross, and Bernd Siebert, Decomposition of degenerate Gromov-Witten invariants, Compos. Math. 156 (2020), no. 10, 2020–2075. MR 4177284, DOI 10.1112/s0010437x20007393
- Dan Abramovich, Qile Chen, Mark Gross, and Bernd Siebert, Punctured logarithmic maps, arXiv:2009.07720 (2021).
- Dan Abramovich and Barbara Fantechi, Orbifold techniques in degeneration formulas, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 (2016), no. 2, 519–579. MR 3559610
- 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
- Younghan Bae and Johannes Schmitt, Chow rings of stacks of prestable curves I, Forum Math. Sigma 10 (2022), Paper No. e28, 47. With an appendix by Bae, Schmitt and Jonathan Skowera MR 4430955, DOI 10.1017/fms.2022.21
- Qile Chen, The degeneration formula for logarithmic expanded degenerations, J. Algebraic Geom. 23 (2014), no. 2, 341–392. MR 3166394, DOI 10.1090/S1056-3911-2013-00614-1
- 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
- William Fulton and Bernd Sturmfels, Intersection theory on toric varieties, Topology 36 (1997), no. 2, 335–353. MR 1415592, DOI 10.1016/0040-9383(96)00016-X
- 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, DOI 10.1007/978-1-4612-1700-8
- 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
- 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
- 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
- Bumsig Kim, Hyenho Lho, and Helge Ruddat, The degeneration formula for stable log maps, Manuscripta Math. 170 (2023), no. 1-2, 63–107. MR 4533479, DOI 10.1007/s00229-021-01361-z
- 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
- 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.
- Sam Molcho, Universal stacky semistable reduction, Israel J. Math. 242 (2021), no. 1, 55–82. MR 4282076, DOI 10.1007/s11856-021-2118-0
- Arthur Ogus, Lectures on logarithmic algebraic geometry, Cambridge Studies in Advanced Mathematics, vol. 178, Cambridge University Press, Cambridge, 2018. MR 3838359, DOI 10.1017/9781316941614
- 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 10.1016/j.ansens.2002.11.001
- Brett Parker, Three dimensional tropical correspondence formula, Comm. Math. Phys. 353 (2017), no. 2, 791–819. MR 3649486, DOI 10.1007/s00220-017-2874-1
- Brett Parker, Tropical gluing formulae for Gromov-Witten invariants, arXiv:1703.05433 (2017).
- Dhruv Ranganathan, Logarithmic Gromov-Witten theory with expansions, Algebr. Geom. 9 (2022), no. 6, 714–761. MR 4518245, DOI 10.14231/AG-2022-022
- The Stacks Project Authors, The Stacks Project, 2020, https://stacks.math.columbia.edu.
- Martin Ulirsch, Functorial tropicalization of logarithmic schemes: the case of constant coefficients, Proc. Lond. Math. Soc. (3) 114 (2017), no. 6, 1081–1113. MR 3661346, DOI 10.1112/plms.12031
- 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
- Jonathan Wang, The moduli stack of $G$-bundles, arXiv:1104.4828 (2011).
- Tony Yue Yu, Enumeration of holomorphic cylinders in log Calabi-Yau surfaces. II: positivity, integrality and the gluing formula, Geom. Topol. 25 (2021), no. 1, 1–46. MR 4226227, DOI 10.2140/gt.2021.25.1
References
- Dan Abramovich, Lucia Caporaso, and Sam Payne, The tropicalization of the moduli space of curves, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 4, 765–809 (English, with English and French summaries). MR 3377065, DOI 10.24033/asens.2258
- Dan Abramovich and Qile Chen, Stable logarithmic maps to Deligne-Faltings pairs, II, Asian J. Math. 18 (2014), no. 3, 465–488.
- Dan Abramovich, Qile Chen, Mark Gross, and Bernd Siebert, Decomposition of degenerate Gromov-Witten invariants, Compos. Math. 156 (2020), no. 10, 2020–2075. MR 4177284, DOI 10.1112/s0010437x20007393
- Dan Abramovich, Qile Chen, Mark Gross, and Bernd Siebert, Punctured logarithmic maps, arXiv:2009.07720 (2021).
- Dan Abramovich and Barbara Fantechi, Orbifold techniques in degeneration formulas, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 (2016), no. 2, 519–579. MR 3559610
- Dan Abramovich, Steffen Marcus, and Jonathan Wise, Comparison theorems for Gromov–Witten invariants of smooth pairs and of degenerations, Annales de l’Institut Fourier 64 (2014), no. 4, 1611–1667 (English). MR 3329675
- Younghan Bae and Johannes Schmitt, Chow rings of stacks of prestable curves I, Forum Math. Sigma 10 (2022), Paper No. e28, 47. With an appendix by Bae, Schmitt and Jonathan Skowera MR 4430955, DOI 10.1017/fms.2022.21
- Qile Chen, The degeneration formula for logarithmic expanded degenerations, J. Algebraic Geom. 23 (2014), no. 2, 341–392. MR 3166394, DOI 10.1090/S1056-3911-2013-00614-1
- 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
- William Fulton and Bernd Sturmfels, Intersection theory on toric varieties, Topology 36 (1997), no. 2, 335–353. MR 1415592, DOI 10.1016/0040-9383(96)00016-X
- 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, DOI 10.1007/978-1-4612-1700-8
- 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
- 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
- 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
- Bumsig Kim, Hyenho Lho, and Helge Ruddat, The degeneration formula for stable log maps, Manuscripta Math. 170 (2023), no. 1-2, 63–107. MR 4533479, DOI 10.1007/s00229-021-01361-z
- 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
- 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.
- Sam Molcho, Universal stacky semistable reduction, Israel J. Math. 242 (2021), no. 1, 55–82. MR 4282076, DOI 10.1007/s11856-021-2118-0
- Arthur Ogus, Lectures on logarithmic algebraic geometry, Cambridge Studies in Advanced Mathematics, vol. 178, Cambridge University Press, Cambridge, 2018. MR 3838359, DOI 10.1017/9781316941614
- 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 10.1016/j.ansens.2002.11.001
- Brett Parker, Three dimensional tropical correspondence formula, Comm. Math. Phys. 353 (2017), no. 2, 791–819. MR 3649486, DOI 10.1007/s00220-017-2874-1
- Brett Parker, Tropical gluing formulae for Gromov-Witten invariants, arXiv:1703.05433 (2017).
- Dhruv Ranganathan, Logarithmic Gromov-Witten theory with expansions, Algebr. Geom. 9 (2022), no. 6, 714–761. MR 4518245
- The Stacks Project Authors, The Stacks Project, 2020, https://stacks.math.columbia.edu.
- Martin Ulirsch, Functorial tropicalization of logarithmic schemes: the case of constant coefficients, Proc. Lond. Math. Soc. (3) 114 (2017), no. 6, 1081–1113. MR 3661346, DOI 10.1112/plms.12031
- 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
- Jonathan Wang, The moduli stack of $G$-bundles, arXiv:1104.4828 (2011).
- Tony Yue Yu, Enumeration of holomorphic cylinders in log Calabi-Yau surfaces. II: Positivity, integrality and the gluing formula, Geom. Topol. 25 (2021), no. 1, 1–46. MR 4226227, DOI 10.2140/gt.2021.25.1
Additional Information
Yixian Wu
Affiliation:
Department of Mathematics, The University of Texas at Austin, 2515 Speedway, Austin, Texas 78712
ORCID:
0000-0002-2856-986X
Email:
yixianwumath@gmail.com
Received by editor(s):
July 13, 2021
Received by editor(s) in revised form:
May 17, 2022
Published electronically:
November 13, 2023
Additional Notes:
This work was supported by NSF grant DMS-1903437.
Article copyright:
© Copyright 2023
University Press, Inc.
