Strengthening the cohomological crepant resolution conjecture for Hilbert–Chow morphisms

Abstract

Given any smooth toric surface S, we prove a SYM-HILB correspondence which relates the 3-point, degree zero, extended Gromov-Witten invariants of the n-fold symmetric product stack [Sym n (S)] of S to the 3-point extremal Gromov-Witten invariants of the Hilbert scheme Hilb n (S) of n points on S. As we do not specialize the values of the quantum parameters involved, this result proves a strengthening of Ruan's Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism Hilb n (S) → Sym n (S) and yields a method of reconstructing the cup product for Hilb n (S) from the orbifold invariants of [Sym n (S)].

Orbifold Gromov–Witten theory of the symmetric product of 𝒜r

“…Let us sketch a basis, constructed in Cheong [8], for the equivariant Chen-Ruan cohomology of the stack OESym n .X /. Indeed, any cohomology-weighted partition .E Á/ with Á i 's cohomology classes on X defines a class on the sector X.…”

“…By virtual localization, (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) can be expressed as a sum of residue integrals over T -fixed components. By Lemma 3.1, the invariant (3-12) is˛.t 1 C t 2 / for some rational number˛, and it suffices to evaluate (3-12) over all T -fixed components of the elements in the union`i ;j F E…”

“…Here D .r C 1/ 2 t 2 1 , and ".F /'s are as in (3-13), (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) respectively.…”

“…where w k is the T -weight of the tangent space to c.D/ at the point c.P k /, and k is the class associated to T c.P k / C k (see (3)(4)(5)(6)(7)(8)(9)(10)). By property (e) in Section 3.…”

“…Theorem 3.9 and Theorem 3.14 provide an effective method to compute 2-point extended invariants of OESym n .A r / in nonzero degrees. With the equations in the following proposition, this also determines the divisor operators as a consequence of 3-point extended invariants in degree zero being determined by the Gromov-Witten theory of OESym n .C 2 /; see Cheong [8]. The domains are nodal curves of genus g and are allowed to be disconnected.…”