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Stringy K-theory and the Chern character | | Tyler J. Jarvis
; Ralph Kaufmann
; Takashi Kimura
; | | Date: |
14 Feb 2005 | | Subject: | Algebraic Geometry; Quantum Algebra; Differential Geometry; K-Theory and Homology MSC-class: 14N35; 53D45 | math.AG math.DG math.KT math.QA | | Abstract: | We define a new orbifold K-theory for a Deligne-Mumford stack; and for a variety with an action of a finite group G we define a G-Frobenius algebra, called the stringy K-theory, whose associated Frobenius algebra of coinvariants is the orbifold K-theory of the quotient stack. The orbifold K-theory is linearly isomorphic to that of Adem-Ruan, but carries a different, "quantum," product. We prove that the stringy,and orbifold, Chern characters are ring isomorphisms from stringy, and orbifold, K-theory to stringy, and orbifold, cohomology, respectively, and that characters respect all properties of a G-Frobenius (respectively Frobenius) algebra that do not involve the metric, and that Grothendieck-Riemann-Roch holds for étale maps. Our main result is a new, simple formula for the obstruction bundle, which allows one to completely exorcise complex curves from the definitions of the stringy Chow ring, stringy K-theory, orbifold K-theory, and Chen-Ruan orbifold cohomology. This new formula plays a key role in the proof that the stringy Chern character is a ring homomorphism and it also yields a simple proof of associativity and the trace axiom. We conclude by showing that a K-theoretic version of Ruan’s conjectures holds for the symmetric product of a complex projective surface with trivial first Chern class. | | Source: | arXiv, math.AG/0502280 | | Services: | Forum | Review | PDF | Favorites | |
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