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Localizing virtual structure sheaves for almost perfect obstruction theories

Part of: Cycles and subschemes Projective and enumerative geometry Families, fibrations

Published online by Cambridge University Press:  07 December 2020

Young-Hoon Kiem  and
Michail Savvas
Young-Hoon Kiem
Affiliation:
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul08826, Korea; E-mail: [email protected]
Michail Savvas
Affiliation:
Department of Mathematics, University of California, San Diego, La Jolla CA92093, USA; E-mail: [email protected]

Abstract

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Almost perfect obstruction theories were introduced in an earlier paper by the authors as the appropriate notion in order to define virtual structure sheaves and K-theoretic invariants for many moduli stacks of interest, including K-theoretic Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds. The construction of virtual structure sheaves is based on the K-theory and Gysin maps of sheaf stacks.

In this paper, we generalize the virtual torus localization and cosection localization formulas and their combination to the setting of almost perfect obstruction theory. To this end, we further investigate the K-theory of sheaf stacks and its functoriality properties. As applications of the localization formulas, we establish a K-theoretic wall-crossing formula for simple $\mathbb{C} ^\ast $ -wall crossings and define K-theoretic invariants refining the Jiang-Thomas virtual signed Euler characteristics.

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants
Secondary: 14C35: Applications of methods of algebraic $K$-theory 14D23: Stacks and moduli problems
Type
Mathematical Physics
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e61
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

References

Alper, Jarod, Hall, Jack and Rydh, David. A Luna étale slice theorem for algebraic stacks. Ann. of Math. (2), 191(3):675–738, 2020.CrossRefGoogle Scholar
Behrend, K.. Gromov-Witten invariants in algebraic geometry. Invent. Math., 127(3):601–617, 1997.CrossRefGoogle Scholar
Behrend, Kai. Donaldson-Thomas type invariants via microlocal geometry. Ann. of Math. (2), 170(3):1307–1338, 2009.CrossRefGoogle Scholar
Behrend, Kai and Fantechi, Barbara. The intrinsic normal cone. Invent. Math., 128(1):45–88, 1997.CrossRefGoogle Scholar
Chang, Huai-Liang, Kiem, Young-Hoon and Li, Jun. Torus localization and wall crossing for cosection localized virtual cycles. Adv. Math., 308:964–986, 2017.CrossRefGoogle Scholar
Edidin, Dan and Graham, William. Nonabelian localization in equivariant $K$ -theory and Riemann-Roch for quotients. Adv. Math., 198(2):547–582, 2005.CrossRefGoogle Scholar
Graber, Tom and Pandharipande, Rahul. Localization of virtual classes. Invent. Math., 135(2):487–518, 1999.CrossRefGoogle Scholar
Hartshorne, Robin. Algebraic geometry . Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52.CrossRefGoogle Scholar
Inaba, Michi-aki. Toward a definition of moduli of complexes of coherent sheaves on a projective scheme. J. Math. Kyoto Univ., 42(2):317–329, 2002.CrossRefGoogle Scholar
Jiang, Yunfeng and Thomas, Richard P.. Virtual signed Euler characteristics. J. Algebraic Geom., 26(2):379–397, 2017.CrossRefGoogle Scholar
Kiem, Young-Hoon. Localizing virtual fundamental cycles for semi-perfect obstruction theories. Internat. J. Math., 29(4):1850032, 30, 2018.CrossRefGoogle Scholar
Kiem, Young-Hoon and Li, Jun. Localizing virtual cycles by cosections. J. Amer. Math. Soc., 26(4):1025–1050, 2013.CrossRefGoogle Scholar
Kiem, Young-Hoon and Li, Jun. A wall crossing formula of Donaldson-Thomas invariants without Chern-Simons functional. Asian J. Math., 17(1):63–94, 2013.CrossRefGoogle Scholar
Kiem, Young-Hoon and Li, Jun. Localizing virtual structure sheaves by cosections. arXiv e-prints, page arXiv:1705.09458, May 2017.Google Scholar
Kiem, Young-Hoon, Li, Jun and Savvas, Michail. Generalized Donaldson-Thomas Invariants via Kirwan Blowups. arXiv e-prints, page arXiv:1712.02544, December 2017.Google Scholar
Kiem, Young-Hoon and Savvas, Michail. K-theoretic generalized Donaldson–Thomas invariants. International Mathematics Research Notices, 05 2020. rnaa097.CrossRefGoogle Scholar
Kim, Bumsig, Kresch, Andrew and Pantev, Tony. Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee. J. Pure Appl. Algebra, 179(1-2):127–136, 2003.CrossRefGoogle Scholar
Kresch, Andrew. Cycle groups for Artin stacks. Invent. Math., 138(3):495–536, 1999.CrossRefGoogle Scholar
Lee, Yuan-Pin. Quantum $K$ -theory. I. Foundations. Duke Math. J., 121(3):389–424, 2004.CrossRefGoogle Scholar
Li, Jun and Tian, Gang. Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. J. Amer. Math. Soc., 11(1):119–174, 1998.CrossRefGoogle Scholar
Lieblich, Max. Moduli of complexes on a proper morphism. J. Algebraic Geom., 15(1):175–206, 2006.CrossRefGoogle Scholar
Manolache, Cristina. Virtual pull-backs. J. Algebraic Geom., 21(2):201–245, 2012.CrossRefGoogle Scholar
Okounkov, Andrei. Lectures on K-theoretic computations in enumerative geometry. In Geometry of moduli spaces and representation theory , volume 24 of IAS/Park City Math. Ser., pages 251–380. Amer. Math. Soc., Providence, RI, 2017.CrossRefGoogle Scholar
Okounkov, Andrei. Takagi lectures on Donaldson-Thomas theory. Jpn. J. Math., 14(1):67–133, 2019.CrossRefGoogle Scholar
Pandharipande, Rahul and Thomas, Richard P.. Curve counting via stable pairs in the derived category. Invent. Math., 178(2):407–447, 2009.CrossRefGoogle Scholar
Qu, Feng. Virtual pullbacks in $K$ -theory. Ann. Inst. Fourier (Grenoble), 68(4):1609–1641, 2018.CrossRefGoogle Scholar
Romagny, Matthieu. Group actions on stacks and applications. Michigan Math. J., 53(1):209–236, 2005.CrossRefGoogle Scholar
Savvas, Michail. Generalized Donaldson-Thomas Invariants of derived objects via Kirwan blowups. arXiv e-prints, page arXiv:2005.13768, May 2020.Google Scholar
Thomas, Richard P.. A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on $K{\kern-.5pt}3$ fibrations. J. Differential Geom., 54(2):367–438, 2000.CrossRefGoogle Scholar
Thomas, Richard P.. Equivariant $K$ -theory and refined Vafa-Witten invariants. Comm. Math. Phys., 378(2):1451–1500, 2020.CrossRefGoogle Scholar
Toen, B.. Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford. $K$ -Theory , 18(1):33–76, 1999.CrossRefGoogle Scholar