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Motivic multiplicative McKay correspondence for surfaces

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Abstract

We revisit the classical two-dimensional McKay correspondence in two respects: The first one, which is the main point of this work, is that we take into account of the multiplicative structure given by the orbifold product; second, instead of using cohomology, we deal with the Chow motives. More precisely, we prove that for any smooth proper two-dimensional orbifold with projective coarse moduli space, there is an isomorphism of algebra objects, in the category of complex Chow motives, between the motive of the minimal resolution and the orbifold motive. In particular, the complex Chow ring (resp. Grothendieck ring, cohomology ring, topological K-theory) of the minimal resolution is isomorphic to the complex orbifold Chow ring (resp. Grothendieck ring, cohomology ring, topological K-theory) of the orbifold surface. This confirms the two-dimensional Motivic Crepant Resolution Conjecture.

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Author notes

  1. Zhiyu Tian

    Present address: Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, China

Authors and Affiliations

  1. Institut Camille Jordan, Université Claude Bernard Lyon 1, Villeurbanne, France

    Lie Fu

  2. CNRS, Institut Fourier, Université Grenoble Alpes, Grenoble, France

    Zhiyu Tian

Authors
  1. Lie Fu
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  2. Zhiyu Tian
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Correspondence to Lie Fu.

Additional information

Lie Fu is supported by the Agence Nationale de la Recherche (ANR) through ECOVA (ANR-15-CE40-0002) and LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon.

Zhiyu Tian is partially supported by the funding “Accueil des Nouveaux Arrivants” of IDEX, Université Grenoble Alpes.

Lie Fu and Zhiyu Tian are supported by ANR through HodgeFun (ANR-16-CE40-0011), and by Projet Exploratoire Premier Soutien (PEPS) Jeunes chercheur-e-s 2016 operated by Insmi and Projet Inter-Laboratoire 2016 and 2017 by Fédération de Recherche en Mathématiques Rhône-Alpes/Auvergne CNRS 3490.

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Fu, L., Tian, Z. Motivic multiplicative McKay correspondence for surfaces. manuscripta math. 158, 295–316 (2019). https://doi.org/10.1007/s00229-018-1026-z

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  • DOI: https://doi.org/10.1007/s00229-018-1026-z

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