Hilbert-Chow Crepant Resolution Conjecture

• The Hilbert-Chow Crepant Resolution Conjecture is a framework connecting orbifold Gromov-Witten invariants of singular symmetric products with the relative invariants of their crepant Hilbert scheme resolutions.
• It employs a universal transformation, often constructed via the Nakajima basis and capped vertex techniques, to align descendant insertions and curve-counting invariants.
• The conjecture extends the GW/DT/Pairs correspondence, offering a modular approach to enumerate curves in complex geometries and under degenerations.

The Hilbert-Chow Crepant Resolution Conjecture concerns the relationship between the orbifold Gromov-Witten theory of symmetric products of surfaces and the Gromov-Witten theory of their crepant resolutions, typically given by Hilbert schemes of points. It posits a deep equivalence of virtual curve-counting invariants, extending the geometric and enumerative understanding of singularities, moduli spaces, and their resolutions, especially within the context of the GW/DT correspondence and its modern enhancements.

1. Background and Conceptual Basis

Let $S$ be a nonsingular surface, and let $\mathrm{Sym}^n(S)$ denote its $n$th symmetric product, realized as the quotient $S^n / \mathfrak{S}_n$. $\mathrm{Sym}^n(S)$ possesses singularities, and its canonical crepant resolution is provided by the Hilbert scheme $\mathrm{Hilb}^n(S)$, parameterizing length $n$ zero-dimensional subschemes of $S$.

The Hilbert-Chow map $\pi: \mathrm{Hilb}^n(S) \to \mathrm{Sym}^n(S)$ realizes this resolution. The conjecture asserts that the Gromov-Witten theory (and other curve or sheaf-counting theories) of the orbifold $[\mathrm{Sym}^n(S)]$ can be matched, via a universal transformation, to the Gromov-Witten theory of $\mathrm{Hilb}^n(S)$, particularly when descendant insertions and boundary conditions are considered. This matching is typically “crepant” (canonical bundle preserved) and is expected to hold at the level of virtual fundamental classes and generating functions.

2. Crepant Resolution Conjecture: Formulation and Descendant Correspondence

The GW/DT/MNOP crepant resolution conjecture claims that for a crepant resolution $\pi: Z \to X$ of a Gorenstein singular variety $X$, the (orbifold) Gromov-Witten theory of $X$ is equivalent, after suitable change of variables and transformation, to the Gromov-Witten theory of $Z$.

In the Hilbert-Chow setting, this takes the form:

• Orbifold GW invariants of $[\mathrm{Sym}^n(\mathbb{C}^2)]$ are universally transformable (in terms of generating functions and descendant insertions) to relative GW invariants of $\mathrm{Hilb}^n(\mathbb{C}^2)$.
• The universal transformation is often constructed via the Nakajima–Grojnowski basis and encoded by correspondence matrices such as $\widetilde{\mathsf K}$, constructed from “capped vertex” computations (Maulik et al., 2023, Pandharipande et al., 2012, Pandharipande, 26 Jan 2025).

For descendant insertions and relative conditions, the correspondence matches:

$$Z^{GW}\big([\mathrm{Sym}^n(S)],u\,|\,\text{orbifold insertions}\big) \quad\longleftrightarrow\quad Z^{GW}\big(\mathrm{Hilb}^n(S),u\,|\,\text{relative conditions}\big),$$

with explicit combinatorial translation governed by orbifold sectors and the Nakajima basis matching Hilbert scheme strata (Pandharipande et al., 2012, Maulik et al., 2023).

3. GW/DT/Pairs Correspondence for Hilbert Schemes and Symmetric Products

The conjecture situates itself as a special case of the general GW/PT/DT descendent correspondence for families of 3-folds and their relative geometries. Relative Gromov-Witten theory of $(X,D)$ (with a divisor $D$) and its sheaf-counting theory (moduli of stable pairs or ideal sheaves) are related via explicit change of variables and correction factors arising from virtual Chern degrees (Pandharipande, 26 Jan 2025).

For the universal family $C^2 \times C \to \overline{M}_{g,n}$, Pandharipande–Tseng prove the GW/PT correspondence with arbitrary relative boundary, which is equivalent to the Crepant Resolution Conjecture for $\mathrm{Hilb}^n(\mathbb{C}^2)$ versus $\mathrm{Sym}^n(\mathbb{C}^2)$ (Pandharipande, 26 Jan 2025):

$$Z^{GW}_{\mathrm{Sym}^n(\mathbb{C}^2)} = Z^{GW}_{\mathrm{Hilb}^n(\mathbb{C}^2)}$$

after universal modification of insertions via the $\widetilde{\mathsf K}$-matrix (explicitly constructed from capped vertex calculations) (Pandharipande et al., 2012).

4. Degenerations, Relative/Logarithmic Structures, and Nakajima Basis

The logarithmic enhancement extends the scope of the conjecture from smooth settings to snc degenerations, gluing formulas, and stratified geometries (Maulik et al., 2023). Central to the matching is the logarithmic Nakajima basis on the cohomology of $\mathrm{Hilb}^n(S)$, accommodating boundary conditions and tangency constraints.

In the log case, one replaces $S$ by its stack of expansions and matches orbifold tangency profiles (partitions $\bm\mu$ and cohomology class $\delta$ on boundary) with Hilbert scheme strata via log cohomology isomorphisms (Maulik et al., 2023):

$$H_{\log}^*(\mathrm{Hilb}^n(S)) \cong \bigoplus_{|\bm\mu|=n} H^*_{\log}\bigg(\prod_{i=1}^r D_i^{\times \ell(\mu_i)}/\mathrm{Aut}(\bm\mu)\bigg)$$

This framework permits strata-by-strata calculation and demonstrates that verifying the correspondence for toric or bundle pieces implies it for general smooth fibers.

5. Master Degeneration Formulas and Exotic Insertions

The universal degeneration formula reflects that global invariants for $Y_\eta$ (the general fiber) are determined by local invariants of strata $X_v$ in the special fiber $Y_0$. The log version strengthens the classical result, showing strata-wise correspondence implies global correspondence (Maulik et al., 2023):

$$\left\langle \prod_i \tau_{a_i}(\zeta_i) \right\rangle_{Y_\eta}^{DT,\beta,\chi} = \sum_\gamma \frac{(-1)^{|\bm\mu|} m_{\bm\mu}}{|\mathrm{Aut}(\bm\mu)|} \prod_v \left\langle \prod_{i\in S_v} \tau_{a_i}(\zeta_i) | \bm\mu_v(\delta_v^{(j)}) \right\rangle^{DT,X_v}_{\gamma(v)}$$

A key new ingredient is the appearance of “exotic” insertions: cohomology classes on log-blowups coupling multiple boundary strata, capturing non-local incidence conditions imposed by singular geometry.

Commutativity diagrams (Theorem 7.25 in (Maulik et al., 2023)) encode that degeneration and correspondence transformations are compatible operations, formalizing the universality and modularity of the crepant resolution conjecture in families and degenerations.

6. Proven Cases, Limitations, and Family Generalizations

The conjecture is established in specific settings:

• Toric threefolds and relative situations, using capped localization and vertex operator formalism (Pandharipande et al., 2012, Oblomkov et al., 2018)
• For families of quintic Calabi-Yau threefolds and $\mathbb{P}^3$ with boundary, via degeneration to unions of toric pieces (Pandharipande, 26 Jan 2025)
• For universal curves: correspondence between Gromov-Witten theory of $\mathrm{Hilb}^n(\mathbb{C}^2)$ and orbifold theory of $\mathrm{Sym}^n(\mathbb{C}^2)$ (Pandharipande, 26 Jan 2025), establishing the Hilbert-Chow crepant resolution case
• The log correspondence and master formulas are proved for snc degenerations, showing the “strata-wise implies global” direction (Maulik et al., 2023)

The general case for arbitrary smooth surfaces, non-toric higher dimensions, and non-stationary descendants remains conjectural, constrained by requirements of rationality and compatibility with virtual classes.

7. Technical Tools and Universal Structures

Critical tools underlying the conjecture include:

• Capped descendent vertex: explicit local calculation matching boundary conditions, forming the universal transformation matrix $\widetilde{\mathsf K}$ (Pandharipande et al., 2012)
• Logarithmic Nakajima basis: encoding Hilbert scheme strata with boundary, facilitating matching of tangencies and insertions (Maulik et al., 2023)
• Degeneration/rubber calculus: gluing local pieces via rubber invariants and the master formula (Pandharipande, 26 Jan 2025, Maulik et al., 2023)
• Operator formalism (vertex operators, Heisenberg algebra, Fock spaces): packaging descendants and their correspondence (Oblomkov et al., 2018)
• Rationality constraints: guaranteeing well-defined analytic change of variables and series expansions in the correspondence (Pandharipande et al., 2012)

The synthesis of these methods reflects the sophistication required to prove equivalence of enumerative invariants under crepant resolution, and situates the Hilbert-Chow conjecture as a core instance within the broader GW/DT/Pairs correspondence landscape.