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We study the relationship between Gromov-Witten invariants of local$\mathbb{P}^4$ and Gromov-witten invariants of $[\mathbb{C}^5/\mathbb{Z}_5]$for all genera. We state the crepant resolution conjecture in explicit form andprove this conjecture for $g=2,3.$

Crepant resolution conjecture for $\mathbb{C}^5/\mathbb{Z}_5$