Stringy K-theory and the Chern character

• 成果类型: Article
• 署名作者: Jarvis, Tyler J.; Kaufmann, Ralph; Kimura, Takashi
• 署名单位: Brigham Young University; University of Connecticut; Boston University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-006-0026-x
• 发表日期: 2007
• 页码: 23-81
• 关键词: orbifold cohomology

摘要

We construct two new G-equivariant rings: K(X, G), called the stringy K-theory of the G-variety X, and H (X,G), called the stringy cohomology of the G-variety X, for any smooth, projective variety X with an action of a finite group G. For a smooth Deligne-Mumford stack, we also construct a new ring K-orb(X) called the full orbifold K-theory of X. We show that for a global quotient X = [X/G], the ring of G-invariants K-orb(X) of K (X, G) is a subalgebra of K-orb([X/G]) and is linearly isomorphic to the orbifold K-theory of Adem-Ruan [AR] (and hence Atiyah-Segal), but carries a different quantum product which respects the natural group grading. We prove that there is a ring isomorphism Ch : K(X, G) -> H(X, G), which we call the stringy Chern character. We also show that there is a ring homomorphism Ch(orb) : K-orb(X) -> H-orb(center dot) (X), which we call the orbifold Chern character, which induces an isomorphism Ch(orb) : K-orb(X) -> H-orb(center dot)(X) when restricted to the sub-algebra K-orb(X). Here H-orb(center dot) is the Chen-Ruan orbifold cohomology. We further show that Ch and Ch(orb) preserve many properties of these algebras and satisfy the Grothendieck-Riemann-Roch theorem with respect to etale maps. All of these results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds. We further prove that H(X, G) is isomorphic to Fantechi and Gottsche's construction [FG,JKK]. Since our constructions do not use complex curves, stable maps, admissible covers, or moduli spaces, our results greatly simplify the definitions of the Fantechi-Gottsche ring, Chen-Ruan orbifold cohomology, and the Abramovich-Graber-Vistoli orbifold Chow ring. We conclude by showing that a K-theoretic version of Ruan's Hyper-Kahler Resolution Conjecture holds for the symmetric product of a complex projective surface with trivial first Chern class.