The Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphisms

TL;DR

This paper proves Ruan's Cohomological Crepant Resolution Conjecture for Hilbert-Chow morphisms using representation theory and universality of Gromov-Witten invariants, reducing the problem to known cases for toric surfaces.

Contribution

It establishes the conjecture for Hilbert schemes by combining vertex operator techniques and universality structures, extending previous results to a broader class of surfaces.

Findings

Proof of Ruan's Conjecture for Hilbert-Chow morphisms.

Reduction of the problem to smooth projective toric surfaces.

Development of universality structures for Gromov-Witten invariants.

Abstract

In this paper, we prove that Ruan's Cohomological Crepant Resolution Conjecture holds for the Hilbert-Chow morphisms. There are two main ideas in the proof. The first one is to use the representation theoretic approach proposed in [QW] which involves vertex operator techniques. The second is to prove certain universality structures about the 3-pointed genus-0 extremal Gromov-Witten invariants of the Hilbert schemes by using the indexing techniques from [LiJ], the product formula from [Beh2] and the co-section localization from [KL1, KL2, LL]. We then reduce Ruan's Conjecture from the case of an arbitrary surface to the case of smooth projective toric surfaces which has already been proved in [Che].