Stringy and Orbiforld Cohomology of Wreath Product Orbifolds

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 n be the symmetric group on n letters. Their semidirect product Gn ⋊ n is called the wreath product of G and it naturally acts on the n-fold product X n, yielding the orbifold [X n/(Gn⋊n)]. Let H (X n,Gn⋊n) be the stringy cohomology [7, 10] of the (Gn⋊n)-space X n. We prove that the space Gn-invariants of H (X n,Gn ⋊ n) is isomorphic to the algebra Hor b([X/G]){n} introduced by Lehn and Sorger [14], where Hor b([X/G]) is the Chen-Ruan 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 X n/(Gn ⋊ n). Furthermore, if H∗(Y ) is isomorphic to Hor b([X/G]) as Frobenius algebras, then H∗(Y [n]) is isomorphic to H∗ or b([X n/(Gn ⋊ n)]) as rings. Thus we verify a special case of the cohomological hyper-Kähler resolution conjecture due to Ruan [22]. 2010 Mathematics Subject Classifications: 14N35 14A20 14E15 14J81