| Let [X/G] be an orbifold which is a global quotient of a compact almost complex manifold X by a finite group G. Let Sigman be the symmetric group on n letters. Their semidirect product Gn ⋊ Sigman is called the wreath product of G and it naturally acts on the n-fold product Xn, yielding the orbifold [Xn/(Gn ⋊ Sigman)]. Let H (Xn, Gn ⋊ Sigman) be the stringy cohomology [8, 12] of the (Gn ⋊ Sigman)-space Xn. When G is Abelian, we show that the algebra of G n-coinvariants of H (Xn, Gn. ⋊ Sigman) is isomorphic to the algebra A {lcub}Sigman{rcub} introduced by Lehn and Sorger [15], where A is the orbifold cohomology of [X/G]. We also prove that, if X is a projective surface with trivial canonical class and Y is a crepant resolution of X/G, then the Hilbert scheme of n points on Y, denoted by Y[n ], is a crepant resolution of Xn/( Gn ⋊ Sigman). Furthermore, if H*( Y) is isomorphic to H*orb ([X/G]), then H*(Y [n]) is isomorphic to H*orb ([Xn] (Gn ⋊ Sigman)]). Thus we verify a special case of the cohomological hyper-Kahler resolution conjecture due to Ruan [24]. |
