Motivic multiplicative McKay correspondence for surfaces

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 pre...

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Bibliographic Details
Published inManuscripta mathematica Vol. 158; no. 3-4; pp. 295 - 316
Main Authors Fu, Lie, Tian, Zhiyu
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2019
Springer Nature B.V
Springer Verlag
Subjects
Algebraic Geometry
Calculus of Variations and Optimal Control; Optimization
Geometry
Homology
Isomorphism
Lie Groups
Mathematics
Mathematics and Statistics
Number Theory
Rings (mathematics)
Topological Groups
55N32
14E16
14J17
14C15
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Summary: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 .
ISSN:0025-2611
1432-1785
DOI:10.1007/s00229-018-1026-z