Home Quantum singularity theory via cosection localization
Article
Licensed
Unlicensed Requires Authentication

Quantum singularity theory via cosection localization

  • Young-Hoon Kiem ORCID logo and Jun Li
Published/Copyright: August 14, 2019

Abstract

We generalize the cosection localized Gysin map to intersection homology and Borel–Moore homology, which provides us with a purely topological construction of the Fan–Jarvis–Ruan–Witten invariants and some GLSM invariants.

Award Identifier / Grant number: DMS-1564500

Award Identifier / Grant number: DMS-1601211

Funding statement: Young-Hoon Kiem was partially supported by Samsung Science and Technology Foundation SSTF-BA1601-01; Jun Li was paritally supported by NSF grants DMS-1564500 and DMS-1601211.

Acknowledgements

We thank Huai-Liang Chang for extended discussions that provided impetus to developing the current theory. We also thank Jinwon Choi, Wei-Ping Li and Yongbin Ruan for useful discussions.

References

[1] A. A. Beĭlinson, J. Bernstein and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces. I (Luminy 1981), Astérisque 100, Société Mathématique de France, Paris (1982), 5–171. Search in Google Scholar

[2] A. Borel, Intersection cohomology, Notes on the seminar held at the University of Bern, Bern, 1983. Search in Google Scholar

[3] G. E. Bredon, Sheaf theory, 2nd ed., Grad. Texts in Math. 170, Springer, New York 1997. 10.1007/978-1-4612-0647-7Search in Google Scholar

[4] H.-L. Chang, Y.-H. Kiem and J. Li, Torus localization and wall crossing for cosection localized virtual cycles, Adv. Math. 308 (2017), 964–986. 10.1016/j.aim.2016.12.019Search in Google Scholar

[5] H.-L. Chang, Y.-H. Kiem and J. Li, Algebraic virtual cycles for quantum singularity theories, preprint (2018), https://arxiv.org/abs/1806.00216; to appear in Comm. Anal. Geom. 10.4310/CAG.2021.v29.n8.a1Search in Google Scholar

[6] H.-L. Chang and J. Li, Gromov–Witten invariants of stable maps with fields, Int. Math. Res. Not. IMRN 2012 (2012), no. 18, 4163–4217. 10.1093/imrn/rnr186Search in Google Scholar

[7] H.-L. Chang, J. Li and W.-P. Li, Witten’s top Chern class via cosection localization, Invent. Math. 200 (2015), no. 3, 1015–1063. 10.1007/s00222-014-0549-5Search in Google Scholar

[8] A. Chiodo, The Witten top Chern class via K-theory, J. Algebraic Geom. 15 (2006), no. 4, 681–707. 10.1090/S1056-3911-06-00444-9Search in Google Scholar

[9] I. Ciocan-Fontanine, D. Favero, J. Guere, B. Kim and M. Shoemaker, Fundamental factorization of a GLSM, Part 1: Construction, preprint (2018), https://arxiv.org/abs/1802.05247. Search in Google Scholar

[10] H. Fan, T. Jarvis and Y. Ruan, The Witten equation, mirror symmetry, and quantum singularity theory, Ann. of Math. (2) 178 (2013), no. 1, 1–106. 10.4007/annals.2013.178.1.1Search in Google Scholar

[11] H. Fan, T. J. Jarvis and Y. Ruan, A mathematical theory of the gauged linear sigma model, Geom. Topol. 22 (2018), no. 1, 235–303. 10.2140/gt.2018.22.235Search in Google Scholar

[12] W. Fulton, Young tableaux. With applications to representation theory and geometry, London Math. Soc. Stud. Texts 35, Cambridge University Press, Cambridge 1997. 10.1017/CBO9780511626241Search in Google Scholar

[13] W. Fulton, Intersection theory, 2nd ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin 1998. 10.1007/978-1-4612-1700-8Search in Google Scholar

[14] M. Goresky and R. MacPherson, Intersection homology theory, Topology 19 (1980), no. 2, 135–162. 10.1016/0040-9383(80)90003-8Search in Google Scholar

[15] M. Goresky and R. MacPherson, Intersection homology. II, Invent. Math. 72 (1983), no. 1, 77–129. 10.1007/BF01389130Search in Google Scholar

[16] M. Goresky and R. MacPherson, Lefschetz fixed point theorem for intersection homology, Comment. Math. Helv. 60 (1985), no. 3, 366–391. 10.1007/978-0-8176-4765-0_8Search in Google Scholar

[17] B. Iversen, Cohomology of sheaves, Universitext, Springer, Berlin 1986. 10.1007/978-3-642-82783-9Search in Google Scholar

[18] Y.-H. Kiem, Intersection cohomology of quotients of nonsingular varieties, Invent. Math. 155 (2004), no. 1, 163–202. 10.1007/s00222-003-0317-4Search in Google Scholar

[19] Y.-H. Kiem, Localizing virtual fundamental cycles for semi-perfect obstruction theories, Internat. J. Math. 29 (2018), no. 4, Article ID 1850032. 10.1142/S0129167X18500325Search in Google Scholar

[20] Y.-H. Kiem and J. Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26 (2013), no. 4, 1025–1050. 10.1090/S0894-0347-2013-00768-7Search in Google Scholar

[21] Y.-H. Kiem and J. Li, Localizing virtual structure sheaves by cosections, preprint (2017), https://arxiv.org/abs/1705.09458; to appear in Int. Math. Res. Not. IMRN. 10.1093/imrn/rny235Search in Google Scholar

[22] F. Kirwan and J. Woolf, An introduction to intersection homology theory, 2nd ed., Chapman & Hall/CRC, Boca Raton 2006. 10.1201/9780367800840Search in Google Scholar

[23] D. B. Massey, The Sebastiani–Thom isomorphism in the derived category, Compos. Math. 125 (2001), no. 3, 353–362. 10.1023/A:1002608716514Search in Google Scholar

[24] A. Polishchuk and A. Vaintrob, Algebraic construction of Witten’s top Chern class, Advances in algebraic geometry motivated by physics (Lowell 2000), Contemp. Math. 276, American Mathematical Society, Providence (2001), 229–249. 10.1090/conm/276/04523Search in Google Scholar

[25] A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, J. reine angew. Math. 714 (2016), 1–122. 10.1515/crelle-2014-0024Search in Google Scholar

[26] C. Voisin, Hodge theory and complex algebraic geometry. II, Cambridge Stud. Adv. Math. 77, Cambridge University, Cambridge 2007. Search in Google Scholar

[27] E. Witten, Algebraic geometry associated with matrix models of two-dimensional gravity, Topological methods in modern mathematics (Stony Brook 1991), Publish or Perish, Houston (1993), 235–269. Search in Google Scholar

Received: 2018-07-24
Revised: 2019-04-23
Published Online: 2019-08-14
Published in Print: 2020-09-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 1.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/crelle-2019-0018/html
Scroll to top button