2005/10/25 by Fabio Perroni, Perroni, Fabio · 1 citation
Mathematics · #14E15 #14N35 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.math/0510528
openalex publication_date 2005/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study Ruan's "cohomological crepant resolution conjecture" (see math.AG/0108195) for orbifolds with transversal ADE singularities. Let [Y] be such an orbifold, Y its coarse moduli space and Z the crepant resolution of Y. Following Ruan [math.AG/0108195], we have a deformation of the cohomology ring H^*(Z) using some Gromov-Witten invariants of Z. The resulting family of rings will be denoted by H^*(Z)(q1,...,qn), where q1,...,qn are complex parameters. In the An case we compute both the orbifold cohomology ring H^*orb([Y]) and the family H^*(Z)(q1,...,qn). The former is achieved in general, the later up to an explicit conjecture on the Gromov-Witten invariants which is verified under additional, technical assumptions. We construct an explicit isomorphism between H^*orb([Y]) and H^*(Z)(-1) in the A1 case, verifying Ruan's conjecture. In the A2 case, the family H^*(q1,q2) is not defined for q1=q2=-1, so the conjecture should be slightly modified. However we give an explicit isomorphism between H^*orb([Y]) and H^*(Z)(q1,q2) with q1=q2 be a primitive third root of the unity, thus proving a slightly modified version of Ruan's conjecture. It is natural to conjecture that, in the An case, H^*orb([Y]) is isomorphic to H^*(Z)(q1,...,qn) with q1=...=qn be a primitive (n+1)-th root of the unity.