Orbifold Cohomology of a Wreath Product Orbifold

Abstract. Let [X/G] be an orbifold which is a global quotient of a compact almost complex manifold X by a finite group G. Let Σn be the symmetric group on n letters. Their semidirect product G ⋊ Σn is called the wreath product of G and it naturally acts on the n-fold product X, yielding the orbifold [X/(G ⋊Σn)]. Let H (X , G ⋊Σn) be the stringy cohomology [FG, JKK1] of the (G ⋊ Σn)-space X . When G is Abelian, we show that the G-coinvariants of H (X, G ⋊ Σn) is isomorphic to the algebra A{Σn} introduced by Lehn and Sorger [LS], 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 , is a crepant resolution of X/(G ⋊ Σn). Furthermore, if H (Y ) is isomorphic to H orb([X/G]), then H(Y ) is isomorphic to H orb([X /(G ⋊ Σn)]). Thus we verify a special case of the cohomological hyper-Kähler resolution conjecture due to Ruan [Ru].