%0 Journal Article %J International Journal of Mathematics. Volume 18, Issue 9, October 2007, Pages 1009-1059 %D 2007 %T Chen-Ruan cohomology of ADE singularities %A Fabio Perroni %K Chen-Ruan cohomology, Ruan\'s conjecture, McKay correspondence %X 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. %B International Journal of Mathematics. Volume 18, Issue 9, October 2007, Pages 1009-1059 %I SISSA %G en %U http://hdl.handle.net/1963/6502 %1 6447 %2 Mathematics %4 1 %# MAT/03 GEOMETRIA %$ Submitted by Fabio Perroni (perroni@sissa.it) on 2013-02-27T13:02:56Z\nNo. of bitstreams: 1\nmath_0605207v2.pdf: 502025 bytes, checksum: c85d5727542636dfdf53276f3ceaf9cd (MD5) %R 10.1142/S0129167X07004436