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LOCALIZATION BY $2$-PERIODIC COMPLEXES AND VIRTUAL STRUCTURE SHEAVES

Part of: Projective and enumerative geometry Cycles and subschemes

Published online by Cambridge University Press:  22 December 2020

Jeongseok Oh  and
Bhamidi Sreedhar Open the ORCID record for Bhamidi Sreedhar [Opens in a new window]
Jeongseok Oh
Affiliation:
School of Mathematics, Korea Institute for Advanced Study (KIAS), 85 Hoegi-ro, Dongdaemun-gu, Seoul02455, Republic of Korea (jeongseok@kias.re.kr, sreedhar@kias.re.kr)
Bhamidi Sreedhar
Affiliation:
School of Mathematics, Korea Institute for Advanced Study (KIAS), 85 Hoegi-ro, Dongdaemun-gu, Seoul02455, Republic of Korea (jeongseok@kias.re.kr, sreedhar@kias.re.kr)
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Abstract

In [12], Kim and the first author proved a result comparing the virtual fundamental classes of the moduli spaces of $\varepsilon $ -stable quasimaps and $\varepsilon $ -stable $LG$ -quasimaps by studying localized Chern characters for $2$ -periodic complexes.

In this paper, we study a K-theoretic analogue of the localized Chern character map and show that for a Koszul $2$ -periodic complex it coincides with the cosection-localized Gysin map of Kiem and Li [11]. As an application, we compare the virtual structure sheaves of the moduli space of $\varepsilon $ -stable quasimaps and $\varepsilon $ -stable $LG$ -quasimaps.

Keywords

virtual structure sheavesmoduli spaces

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
Secondary: 14C35: Applications of methods of algebraic $K$-theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 5 , September 2022 , pp. 1477 - 1506
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Ballard, M., Favero, D. and Katzarkov, L., A category of kernels for equivariant factorizations and its implications for Hodge theory, Publ. Math. Inst. Hautes Études Sci. 120(1) (2014), 1–111.CrossRefGoogle Scholar
Chang, H.-L. and Li, J., Gromov-Witten invariants of stable maps with fields, Int. Math. Res. Not. IMRN 2012(18) (2012), 4163–4217.Google Scholar
Cheong, D., Ciocan-Fontanine, I. and Kim, B., Orbifold quasimap theory, Math. Ann. 363(3-4) (2015), 777–816.CrossRefGoogle Scholar
Chiodo, A., The Witten top Chern class via K-theory, J. Algebraic Geom. 15 (2006), 681–707.CrossRefGoogle Scholar
Ciocan-Fontanine, I., Favero, D., Guéré, J., Kim, B. and Shoemaker, M., ‘Fundamental factorization of GLSM part I: Construction’, Preprint, 2018, arXiv:1802.05247.Google Scholar
Ciocan-Fontanine, I. and Kim, B., ‘Quasimap wall-crossings and mirror symmetry’, Preprint, 2016, arXiv:1611.05023.Google Scholar
Ciocan-Fontanine, I., Kim, B. and Maulik, D., Stable quasimaps to GIT quotients, J. Geom. Phys. 75 (2014), 17–47.CrossRefGoogle Scholar
Fan, H., Jarvis, T. and Ruan, Y., A mathematical theory of the gauged linear sigma model, Geom. Topol. 22(1) (2018), 235–303.CrossRefGoogle Scholar
Hall, J. and Rydh, D., Perfect complexes on algebraic stacks, Compos. Math. 153(11) (2017), 2318–2367.CrossRefGoogle Scholar
Kiem, Y.-H. and Li, J., Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26(4) (2013), 1025–1050.CrossRefGoogle Scholar
Kiem, Y.-H. and Li, J., Localizing virtual structure sheaves by cosections, Int. Math. Res. Not. IMRN (2018), rny235.Google Scholar
Kim, B. and Oh, J., ‘Localized Chern Characters for $2$ -periodic complexes’, Preprint, 2018, arXiv:1804.03774.Google Scholar
Lee, Y.-P., Quantum K-theory I: Foundations, Duke Math. J. 121(3) (2004), 389–424.Google Scholar
Lin, K. H. and Pomerleano, D., Global matrix factorizations, Math. Res. Lett. 20(1) (2013), 91–106.Google Scholar
Nironi, F., ‘Grothendieck duality for Deligne-Mumford stacks’, Preprint, 2008, arXiv:0811.1955.Google Scholar
Orlov, D., Formal completions and idempotent completions of triangulated categories of singularities, Adv. Math. 226(1) (2011), 206–217.Google Scholar
Orlov, D., Matrix factorizations for nonaffine LG–models, Math. Ann. 353(1) (2012), 95–108.CrossRefGoogle Scholar
Polishchuk, A. and Vaintrob, A., Algebraic construction of Wittens top Chern class, in Advances in Algebraic Geometry Motivated by Physics (Lowell, MA, 2000), 229–249 (Providence, American Mathematical Society, 2001).Google Scholar
Polishchuk, A. and Vaintrob, A., Matrix factorizations and singularity categories for stacks, Ann. Inst. Fourier (Grenoble) 61(7) (2011), 2609–2642.Google Scholar
Thomas, R. P., ‘Equivariant K-theory and refined Vafa-Witten invariants’, Preprint, 2018, arXiv:1810.00078.Google Scholar
Toen, B., Théorèmes de Riemann-Roch pour les champs de Deligne–Mumford, $K$ -theory 18(1) (1999), 33–76.Google Scholar
Qu, F., Virtual pullbacks in $K$ -theory, Ann. Inst. Fourier (Grenoble) 68(4) (2018), 1609–1641.CrossRefGoogle Scholar