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ATIYAH CLASS AND CHERN CHARACTER FOR GLOBAL MATRIX FACTORISATIONS

Part of: (Co)homology theory Homological algebra

Published online by Cambridge University Press:  08 January 2021

Bumsig Kim  and
Alexander Polishchuk Open the ORCID record for Alexander Polishchuk [Opens in a new window]
Bumsig Kim
Affiliation:
Korea Institute for Advanced Study
Alexander Polishchuk
Affiliation:
University of Oregon, Korea Institute for Advanced Study and National Research University Higher School of Economics, Russian Federation
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Abstract

We define the Atiyah class for global matrix factorisations and use it to give a formula for the categorical Chern character and the boundary-bulk map for matrix factorisations, generalising the formula in the local case obtained in [12]. Our approach is based on developing the Lie algebra analogies observed by Kapranov [7] and Markarian [9].

Keywords

matrix factorisationAtiyah classcategorical Chern characterboundary-bulk mapHochschild-Kostand-Rosenberg isomorphism

MSC classification

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions
Secondary: 18G60: Other (co)homology theories
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 4 , July 2022 , pp. 1445 - 1470
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Atiyah, M., Complex analytic connections in fibre bundles, Trans. AMS 85 (1957), 181–207.CrossRefGoogle Scholar
Calaque, D. and Van den Bergh, M., Hochschild cohomology and Atiyah classes, Adv. Math. 224 (2010), 1839–1889.CrossRefGoogle Scholar
Căldăraru, A., The Mukai pairing. II: The Hochschild-Kostant-Rosenberg isomorphism, Adv. Math. 194(1) (2005), 34–66.CrossRefGoogle Scholar
Dyckerhoff, T. and Murfet, D., Pushing forward matrix factorizations, Duke Math. J. 162 (2013), 1249–1311.CrossRefGoogle Scholar
Efimov, A. and Positselski, L., Coherent analogues of matrix factorizations and relative singularity categories, Algebra Number Theory 9(3) (2015), 1159–1292.CrossRefGoogle Scholar
Huybrechts, D. and Lehn, M., The Geometry of Moduli Spaces of Sheaves, 2nd ed. (Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010).CrossRefGoogle Scholar
Kapranov, M., Rozansky-Witten invariants via Atiyah classes, Compositio Math. 115(1) (1999), 71–113.CrossRefGoogle Scholar
Lin, K. H. and Pomerleano, D., Global matrix factorizations, Math. Res. Lett. 20(1) (2013), 91–106.CrossRefGoogle Scholar
Markarian, N., The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem, J. Lond. Math. Soc. (2) 79(1) (2009), 129–143.CrossRefGoogle Scholar
Platt, D., Chern character for global matrix factorizations, arXiv:1209.5686.Google Scholar
Polishchuk, A. and Positselski, L., Quadratic Algebras (American Mathematical Society, Providence, RI, 2005).CrossRefGoogle Scholar
Polishchuk, A. and Vaintrob, A., Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations, Duke Math. J. 161(10) (2012), 1863–1926.CrossRefGoogle Scholar
Polishchuk, A. and Vaintrob, A., Matrix factorizations and cohomological field theories, J. Reine Angew. Math. 714 (2016), 1–122.CrossRefGoogle Scholar
Preygel, A., Thom-Sebastiani and duality for matrix factorizations, and results on the higher structures of the Hochschild invariants, arXiv:1101.5834.Google Scholar
Ramadoss, A. C., A generalized Hirzebruch-Riemann-Roch theorem, C. R. Math. Acad. Sci. Paris 347(5–6) (2009), 289–292.CrossRefGoogle Scholar
Roberts, J. and Willerton, S., On the Rozansky-Witten weight systems, Algebr. Geom. Topol. 10(3) (2010), 1455–1519.CrossRefGoogle Scholar
Yekutieli, A., The continuous Hochschild cochain complex of a scheme, Can. J. Math. 54(6) (2002), 1319–1337.CrossRefGoogle Scholar
Yu, X., Chern character for matrix factorizations via Chern-Weil, J. Algebra 424 (2015), 416–447.CrossRefGoogle Scholar