Atiyah-Segal theorem for Deligne-Mumford stacks and applications

Krishna, Amalendu; Sreedhar, Bhamidi

doi:10.1090/jag/755

Authors: Amalendu Krishna and Bhamidi Sreedhar
Journal: J. Algebraic Geom. 29 (2020), 403-470
DOI: https://doi.org/10.1090/jag/755
Published electronically: February 3, 2020
MathSciNet review: 4158458
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Abstract | References | Additional Information

Abstract: We prove an Atiyah-Segal isomorphism for the higher $K$-theory of coherent sheaves on quotient Deligne-Mumford stacks over $\mathbb {C}$. As an application, we prove the Grothendieck-Riemann-Roch theorem for such stacks. This theorem establishes an isomorphism between the higher $K$-theory of coherent sheaves on a Deligne-Mumford stack and the higher Chow groups of its inertia stack. Furthermore, this isomorphism is covariant for proper maps between Deligne-Mumford stacks.

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Additional Information

Amalendu Krishna
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bha-bha Road, Colaba, Mumbai, India
MR Author ID: 703987
Email: amal@math.tifr.res.in

Bhamidi Sreedhar
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Colaba, Mumbai, India
Address at time of publication: KIAS, 85 Hoegi-ro, Dongdaemun-gu, Seoul 02455, Republic of Korea
Email: sreedhar@kias.re.kr

Received by editor(s): October 13, 2017
Received by editor(s) in revised form: July 12, 2019
Published electronically: February 3, 2020
Additional Notes: This work was partly completed when the first author was visiting IMS at the National University of Singapore during the program on Higher Dimensional Algebraic Geometry, Holomorphic Dynamics and Their Interactions in January 2017. He would like to thank the institute and Professor De-Qi Zhang for the invitation and support.
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