TY - JOUR

T1 - Adams operations on the virtual K -theory of P (1, n)

AU - Kimura, Takashi

AU - Sweet, Ross

N1 - Publisher Copyright:
© 2017 World Scientific Publishing Company.

PY - 2017/8/1

Y1 - 2017/8/1

N2 - We analyze the structure of the virtual (orbifold) K-theory ring of the complex orbifold P(1,n) and its virtual Adams (or power) operations, by using the non-Abelian localization theorem of Edidin-Graham [D. Edidin and W. Graham, Nonabelian localization in equivariant K-theory and Riemann-Roch for quotients, Adv. Math. 198(2) (2005) 547-582]. In particular, we identify the group of virtual line elements and obtain a natural presentation for the virtual K-theory ring in terms of these virtual line elements. This yields a surjective homomorphism from the virtual K-theory ring of P(1,n) to the ordinary K-theory ring of a crepant resolution of the cotangent bundle of P(1,n) which respects the Adams operations. Furthermore, there is a natural subring of the virtual K-theory ring of P(1,n) which is isomorphic to the ordinary K-theory ring of the resolution. This generalizes the results of Edidin-Jarvis-Kimura [D. Edidin, T. J. Jarvis and T. Kimura, Chern classes and compatible power operation in inertial K-theory, Ann. K-Theory (2016)], who proved the latter for n = 2, 3.

AB - We analyze the structure of the virtual (orbifold) K-theory ring of the complex orbifold P(1,n) and its virtual Adams (or power) operations, by using the non-Abelian localization theorem of Edidin-Graham [D. Edidin and W. Graham, Nonabelian localization in equivariant K-theory and Riemann-Roch for quotients, Adv. Math. 198(2) (2005) 547-582]. In particular, we identify the group of virtual line elements and obtain a natural presentation for the virtual K-theory ring in terms of these virtual line elements. This yields a surjective homomorphism from the virtual K-theory ring of P(1,n) to the ordinary K-theory ring of a crepant resolution of the cotangent bundle of P(1,n) which respects the Adams operations. Furthermore, there is a natural subring of the virtual K-theory ring of P(1,n) which is isomorphic to the ordinary K-theory ring of the resolution. This generalizes the results of Edidin-Jarvis-Kimura [D. Edidin, T. J. Jarvis and T. Kimura, Chern classes and compatible power operation in inertial K-theory, Ann. K-Theory (2016)], who proved the latter for n = 2, 3.

KW - Adams operation

KW - Equivalent K -theory

KW - Gromov-Witten theory

KW - crepant resolution conjecture

KW - orbifold K -theory

UR - http://www.scopus.com/inward/record.url?scp=84988660463&partnerID=8YFLogxK

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U2 - 10.1142/S0219498817501493

DO - 10.1142/S0219498817501493

M3 - Article

AN - SCOPUS:84988660463

SN - 0219-4988

VL - 16

JO - Journal of Algebra and Its Applications

JF - Journal of Algebra and Its Applications

IS - 8

M1 - 1750149

ER -