Localizing virtual cycles by cosections

Kiem, Young-Hoon; Li, Jun

doi:10.1090/S0894-0347-2013-00768-7

Localizing virtual cycles by cosections
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by Young-Hoon Kiem and Jun Li;

J. Amer. Math. Soc. 26 (2013), 1025-1050

DOI: https://doi.org/10.1090/S0894-0347-2013-00768-7

Published electronically: March 27, 2013

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Abstract:

We show that a cosection of the obstruction sheaf of a perfect obstruction theory localizes the virtual cycle to the non-surjective locus of the cosection. We construct a localized Gysin map and localized virtual cycles. Various applications of this construction are discussed.

References

Kai Behrend, Donaldson-Thomas type invariants via microlocal geometry, Ann. of Math. (2) 170 (2009), no. 3, 1307–1338. MR 2600874, DOI 10.4007/annals.2009.170.1307
K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88. MR 1437495, DOI 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 10.1090/S0894-0347-00-00326-X
Jim Bryan and Naichung Conan Leung, Generating functions for the number of curves on abelian surfaces, Duke Math. J. 99 (1999), no. 2, 311–328. MR 1708022, DOI 10.1215/S0012-7094-99-09911-8
Huai-Liang Chang and Young-Hoon Kiem Poincaré invariants are Seiberg-Witten invariants. To appear in Geometry and Topology, arXiv:1205.0848.
Huai-Liang Chang and Jun Li, Gromov-Witten invariants of stable maps with fields, Int. Math. Res. Not. IMRN 18 (2012), 4163–4217. MR 2975379, DOI 10.1093/imrn/rnr186
Huai-Liang Chang and Jun Li. In preparation.
S. K. Donaldson, Yang-Mills invariants of four-manifolds, Geometry of low-dimensional manifolds, 1 (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 150, Cambridge Univ. Press, Cambridge, 1990, pp. 5–40. MR 1171888
Markus Dürr, Alexandre Kabanov, and Christian Okonek, Poincaré invariants, Topology 46 (2007), no. 3, 225–294. MR 2319736, DOI 10.1016/j.top.2007.02.004
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
C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), no. 1, 173–199. MR 1728879, DOI 10.1007/s002229900028
Young-Hoon Kiem and Jun Li. Gromov-Witten invariants of varieties with holomorphic 2-forms. ArXiv:0707.2986.
Young-Hoon Kiem and Jun Li, Low degree GW invariants of spin surfaces, Pure Appl. Math. Q. 7 (2011), no. 4, Special Issue: In memory of Eckart Viehweg, 1449–1475. MR 2918169, DOI 10.4310/PAMQ.2011.v7.n4.a17
Young-Hoon Kiem and Jun Li, Low degree GW invariants of surfaces II, Sci. China Math. 54 (2011), no. 8, 1679–1706. MR 2824966, DOI 10.1007/s11425-011-4258-x
Young-Hoon Kiem and Jun Li, A wall crossing formula of Donaldson-Thomas invariants without Chern-Simons functional, Asian J. Math. 17 (2013), no. 1, 63–94.
Andrew Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), no. 3, 495–536. MR 1719823, DOI 10.1007/s002220050351
Bumsig Kim, Andrew Kresch, and Tony Pantev, Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee, J. Pure Appl. Algebra 179 (2003), no. 1-2, 127–136. MR 1958379, DOI 10.1016/S0022-4049(02)00293-1
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
Junho Lee, Family Gromov-Witten invariants for Kähler surfaces, Duke Math. J. 123 (2004), no. 1, 209–233. MR 2060027, DOI 10.1215/S0012-7094-04-12317-6
Junho Lee, Holomorphic 2-forms and vanishing theorems for Gromov-Witten invariants, Canad. Math. Bull. 52 (2009), no. 1, 87–94. MR 2494314, DOI 10.4153/CMB-2009-011-1
Junho Lee and Thomas H. Parker, A structure theorem for the Gromov-Witten invariants of Kähler surfaces, J. Differential Geom. 77 (2007), no. 3, 483–513. MR 2362322
Jun Li, A note on enumerating rational curves in a $K3$ surface, Geometry and nonlinear partial differential equations (Hangzhou, 2001) AMS/IP Stud. Adv. Math., vol. 29, Amer. Math. Soc., Providence, RI, 2002, pp. 53–62. MR 1926435, DOI 10.1090/amsip/029/07
Jun Li and Wei-Ping Li, Two point extremal Gromov-Witten invariants of Hilbert schemes of points on surfaces, Math. Ann. 349 (2011), no. 4, 839–869. MR 2777035, DOI 10.1007/s00208-010-0542-2
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, DOI 10.1090/S0894-0347-98-00250-1
Jun Li and Gang Tian, Comparison of algebraic and symplectic Gromov-Witten invariants, Asian J. Math. 3 (1999), no. 3, 689–728. MR 1793677, DOI 10.4310/AJM.1999.v3.n3.a7
John Mather, Notes on topological stability, Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 4, 475–506. MR 2958928, DOI 10.1090/S0273-0979-2012-01383-6
D. Maulik and R. Pandharipande, New calculations in Gromov-Witten theory, Pure Appl. Math. Q. 4 (2008), no. 2, Special Issue: In honor of Fedor Bogomolov., 469–500. MR 2400883, DOI 10.4310/PAMQ.2008.v4.n2.a7
D. Maulik, R. Pandharipande, and R. P. Thomas, Curves on $K3$ surfaces and modular forms, J. Topol. 3 (2010), no. 4, 937–996. With an appendix by A. Pixton. MR 2746343, DOI 10.1112/jtopol/jtq030
A. Okounkov and R. Pandharipande, Quantum cohomology of the Hilbert scheme of points in the plane, Invent. Math. 179 (2010), no. 3, 523–557. MR 2587340, DOI 10.1007/s00222-009-0223-5
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