The cohomological crepant resolution conjecture for the Hilbert-Chow morphisms (Q503074)

The cohomological crepant resolution conjecture of \textit{Y. Ruan} [Contemp. Math. 403, 117--126 (2006; Zbl 1105.14078)] states that if \(f: W \to Z\) is a crepant reolution of an orbifold \(Z\), then the orbifold cohomology ring \(H^*_{\text{CR}}(Z)\) introduced by \textit{W. Chen} and \textit{Y. Ruan} [Commun. Math. Phys. 248, No. 1, 1--31 (2004; Zbl 1063.53091)] is isomorphic to the ring \(H^*_f (W)\) obtained from the ordinary cohomology ring of \(W\) by adding quantum corrections related to curves contracted by \(f\). For a simply connected smooth, projective surface \(X\), the authors prove the conjecture for the Hilbert-Chow morphism \(\rho_n: X^{[n]} \to X^{(n)}\) from the Hilbert scheme of \(n\) points to the symmetric product, extending the previously known special cases \(n=2,3\) [\textit{D. Edidin} et al., Asian J. Math. 7, No. 4, 551--574 (2003; Zbl 1079.14061)] and [\textit{W.-P. Li} and \textit{Z. Qin}, Turk. J. Math. 26, No. 1, 53--68 (2002; Zbl 1054.14068)], \(K_X\) is trivial [\textit{B. Fantechi} and \textit{L. Göttsche}, Duke Math. J. 117, No. 2, 197--227 (2003; Zbl 1086.14046)] and [\textit{M. Lehn} and \textit{C. Sorger}, Invent. Math. 152, No. 2, 305--329 (2003; Zbl 1035.14001)] and \(X\) is toric [\textit{W. K. Cheong}, Math. Ann. 356, No. 1, 45--72 (2013; Zbl 1277.14044)]. The proof uses the axiomatic approach developed by \textit{Z. Qin} and \textit{W. Wang} [Contemp. Math. 310, 233--257 (2002; Zbl 1045.14001)] to reduce the question to checking some relations among Heisenberg operators [\textit{H. Nakajima}, Ann. Math. (2) 145, No. 2, 379--388 (1997; Zbl 0915.14001)] acting on \(\mathbb H_X = \bigoplus_{n=0}^\infty H^*(X^{(n)})\) and on \(\oplus_{n} H^*_{\rho_n} (X^{[n]})\). These relations were known for \(H^*(X^{(n)})\) by work of \textit{Qin and Wang} [loc. cit.] and are proved for \(H^*(X^{[n]})\) when \(X\) is toric. To obtain the general case, the authors are able to reduce to the toric case by proving certain universality structures on the 3-pointed genus-0 Gromov-Witten invariants of \(X^{[n]}\) using the indexing techniques of \textit{J. Li} [Geom. Topology 10, 2117--2171 (2006; Zbl 1140.14012)], \textit{K. Behrend}'s product formula for Gromov-Witten invariants [J. Alg. Geom. 8, 529--541 (1999; Zbl 0938.14032)] and the co-sectional localization techniques of \textit{J. Li} and \textit{W.-P. Li} [Math. Ann. 349, 839--869 (2011; Zbl 1221.14006)] and \textit{Y.-H. Kiem} and \textit{J. Li} [J. Amer. Math. Soc. 26, 1025--1050 (2013; Zbl 1276.14083)].