The cohomological crepant resolution conjecture for the Hilbert–Chow morphisms

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]. Hilbert schemes of points on surfacesLet X be a smooth projective complex surface with the canonical class K X and the Euler class e X , and X [n] be the Hilbert scheme of points in X. An element in X [n] is represented by a length-n 0-dimensional closed subscheme ξ of X. It is well known that X [n] is smooth. For a subset Y ⊂ X, define M n (Y ) = {ξ ∈ X [n] | Supp(ξ) = {x} for some x ∈ Y }.

The Gromov–Witten invariants of the Hilbert schemes of points on surfaces with pg > 0

“…Gromov-Witten invariants of the Hilbert schemes determined; moreover, the quantum cohomology of X [2] coincides with its quantum corrected cohomology [19,20]. Theorem 1.3.…”

“…Using this technique, Kiem and Li [15,16] studied the Gromov-Witten theory of minimal surfaces of general type, and Li and the second author [18] computed the quantum boundary operator for the Hilbert schemes of points on surfaces. Cosection localization also played a pivotal role in [20] determining the structure of genus-0 extremal Gromov-Witten invariants of these Hilbert schemes and verifying the Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphisms.…”

“…It follows that if X is a simply connected surface admitting a holomorphic differential two-form with irreducible zero divisor and satisfying K 2 X > 1, then all the Gromov-Witten invariants (without descendant insertions) of X [2] can be determined; moreover, the quantum cohomology of X [2] coincides with its quantum corrected cohomology [LQ1,LQ2].…”

Extremal Gromov-Witten invariants of the Hilbert scheme of Points

“…Via a representation theoretic approach, [28] presents a complicated proof of Ruan's Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism 𝜌 𝑛 : 𝑋 [𝑛] → 𝑋 (𝑛) for all 𝑛 ≥ 1. As an application of Theorem 4.10 (together with the results in [23,27] about the 1-point and 2-point genus-0 extremal Gromov-Witten invariants of 𝑋 [𝑛] ), we now give a direct (but tedious) proof of this conjecture when 𝑛 = 3.…”

Torus localization and wall crossing for cosection localized virtual cycles

“…. The cosection localized virtual fundamental class turned out to be quite useful ( [2,5,7,9,13,14,16,17,19,20,23,27,29,30,31]). For further applications, it seems desirable to have cosection localized analogues for torus localization formula, virtual pullback and wall crossing formulas.…”