Title | Chen-Ruan cohomology of ADE singularities |

Publication Type | Journal Article |

Year of Publication | 2007 |

Authors | Perroni, F |

Journal | International Journal of Mathematics. Volume 18, Issue 9, October 2007, Pages 1009-1059 |

Keywords | Chen-Ruan cohomology, Ruan\'s conjecture, McKay correspondence |

Abstract | We study Ruan\'s \\textit{cohomological crepant resolution conjecture} for\r\norbifolds with transversal ADE singularities. In the $A_n$-case we compute both\r\nthe Chen-Ruan cohomology ring $H^*_{\\rm CR}([Y])$ and the quantum corrected\r\ncohomology ring $H^*(Z)(q_1,...,q_n)$. The former is achieved in general, the\r\nlater up to some additional, technical assumptions. We construct an explicit\r\nisomorphism between $H^*_{\\rm CR}([Y])$ and $H^*(Z)(-1)$ in the $A_1$-case,\r\nverifying Ruan\'s conjecture. In the $A_n$-case, the family\r\n$H^*(Z)(q_1,...,q_n)$ is not defined for $q_1=...=q_n=-1$. This implies that\r\nthe conjecture should be slightly modified. We propose a new conjecture in the\r\n$A_n$-case which we prove in the $A_2$-case by constructing an explicit\r\nisomorphism. |

URL | http://hdl.handle.net/1963/6502 |

DOI | 10.1142/S0129167X07004436 |

## Chen-Ruan cohomology of ADE singularities

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