Chen-Ruan Cohomology and Stringy Orbifold K-Theory for Stable Almost Complex Orbifolds
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Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and
Springer-Verlag Berlin Heidelberg 2020
Chen-Ruan Cohomology and Stringy Orbifold K-Theory for Stable Almost Complex Orbifolds∗ Chengyong DU1
Tiyao LI2
Abstract Comparing to the construction of stringy cohomology ring of equivariant stable almost complex manifolds and its relation with the Chen-Ruan cohomology ring of the quotient almost complex orbifolds, the authors construct in this note a Chen-Ruan cohomology ring for a stable almost complex orbifold. The authors show that for a finite group G and a G-equivariant stable almost complex manifold X, the G-invariant part of the stringy cohomology ring of (X, G) is isomorphic to the Chen-Ruan cohomology ring of the global quotient stable almost complex orbifold [X/G]. Similar result holds when G is a torus and the action is locally free. Moreover, for a compact presentable stable almost complex orbifold, they study the stringy orbifold K-theory and its relation with Chen-Ruan cohomology ring. Keywords Stable almost complex orbifolds, Chen-Ruan cohomology, Orbifold Ktheory, Stringy product 2000 MR Subject Classification 55N32, 53D45, 55N15, 19L10
1 Introduction Since the introduction of Chen-Ruan cohomology ring (see [9]) of almost complex orbifolds and orbifold Gromov-Witten theory (see [8]) of compact symplectic orbifolds, there are lots of works on related area. The most simple orbifolds are global quotient orbifolds. Let G be a finite group and X be a G-equivariant almost complex manifold, the global quotient orbifold [X/G] is an almost complex orbifold. In 2003, Fantechi-G¨ ottsche [14] constructed ∗ a stringy cohomology ring H (X, G), which they called orbifold cohomology, for the pair (X, G) by following the construction of Chen-Ruan cohomology ring in [9], and showed that H ∗ (X, G)G , the G-invariant part of H ∗ (X, G), is isomorphic to the Chen-Ruan cohomology ∗ ring HCR ([X/G]) as Frobenius algebras. Their construction of stringy cohomology ring works for general G-equivariant stable almost complex manifolds. In 2007, Jarvis-Kaufmann-Kimura [17] constructed the stringy Chow ring and stringy K-theory for G-varieties when G is finite. Manuscript received February 23, 2018. Revised October 8, 2018. of mathematics and V. C. & V. R. Key Lab, Sichuan Normal University, Chengdu 610068, China E-mail: [email protected] 2 School of Mathematics, Chongqing Normal University, Chongqing 401331, China. E-mail: [email protected] ∗ This work was supported by the National Natural Science Foundation of China (Nos. 11501393, 11626050, 11901069), Sichuan Science and Technology Program (No. 2019YJ0509), joint research project of Laurent Mathematics Research Center of Sichuan Normal University and V. C. & V. R. Key Lab of Sichuan Province, by Science and Technology Research Program of Chongqing Education Commission of China (No. KJ1600324) and Natural Science Foundation of Chongqing, China (No. cstc2018jcyjAX0465). 1 School
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C. Y. Du and T. Y. Li
They also constructed a stringy Chern character wh
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