TAILIEUCHUNG - Đề tài " McKay correspondence for elliptic genera "

We establish a correspondence between orbifold and singular elliptic genera of a global quotient. While the former is defined in terms of the fixed point set of the action, the latter is defined in terms of the resolution of singularities. As a byproduct, the second quantization formula of Dijkgraaf, Moore, Verlinde and Verlinde is extended to arbitrary Kawamata log-terminal pairs. 1. Introduction One of the fundamental problems suggested by the intersection homology theory is to determine which characteristic numbers can be defined for singular varieties. . | Annals of Mathematics McKay correspondence for elliptic genera By Lev Borisov and Anatoly Libgober Annals of Mathematics 161 2005 1521 1569 McKay correspondence for elliptic genera By Lev Borisov and ÀNATOLy Libgober Abstract We establish a correspondence between orbifold and singular elliptic genera of a global quotient. While the former is defined in terms of the fixed point set of the action the latter is defined in terms of the resolution of singularities. As a byproduct the second quantization formula of Dijkgraaf Moore Verlinde and Verlinde is extended to arbitrary Kawamata log-terminal pairs. 1. Introduction One of the fundamental problems suggested by the intersection homology theory is to determine which characteristic numbers can be defined for singular varieties. Elliptic genus appears to be a key tool for a solution to this problem. In 30 it was shown that the Chern numbers invariant in small resolutions are determined by the elliptic genus of such a resolution. In 7 the elliptic genus was defined for singular varieties with Q-Gorenstein Kawamata-logterminal singularities and its behavior in resolutions of singularities was studied. Among other things 7 shows that the elliptic genus is invariant in crepant and in particular small resolutions whenever they exist. Hence the elliptic genus for such class of singular varieties provides the complete class of Chern numbers which is possible to define in such singular setting. In present work we study the elliptic genus of singular varieties which are global quotients. We obtain generalizations for several relations between the numerical invariants of actions of finite groups acting on algebraic varieties and invariant of resolutions. Much of the interest in such relations comes from works in physics and the work on Hilbert schemes cf. 12 18 11 16 but starts with the work of McKay 28 . The McKay correspondence was originally proposed in 28 as a relation between minimal resolutions of quotient singularities C2

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