Jun Li's Degeneration Formula in GW Theory

• Jun Li's degeneration formula is a method in enumerative geometry that computes Gromov–Witten invariants by decomposing smooth fiber invariants into sums of relative invariants.
• It utilizes precise gluing of moduli spaces, matching combinatorial splittings with contact orders and automorphism factors to capture global enumerative data.
• Extensions of the formula to orbifold, logarithmic, and analytic frameworks broaden its applicability across algebraic, symplectic, and tropical geometry.

Jun Li’s degeneration formula provides a foundational tool for computing Gromov–Witten invariants of a smooth fiber in a one-parameter family by expressing them in terms of relative (or orbifold-relative) invariants of the components of a singular fiber. Originating in algebraic geometry, the formula has become central in enumerative geometry, symplectic topology, and orbifold Gromov–Witten theory, with extensions to logarithmic and analytic frameworks. The key mechanism is a precise gluing of moduli spaces, capturing global enumerative data via sums over discrete combinatorial splittings, weighted by contact orders and automorphisms.

1. Algebraic Framework and Moduli Theory

The classical context is a flat degeneration $\pi: \mathcal{X} \to \Delta$, where $\Delta$ is a disk, the general fiber $X_t$ for $t\neq0$ is smooth, and the special fiber $X_0 = X_1 \cup_D X_2$ consists of two smooth varieties intersecting transversely along a smooth divisor $D$. The moduli spaces in question are:

• $\overline{M}_{g,n}(X_t, \beta)$: Stable maps of genus $g$, $n$ marked points and class $\beta$ to the smooth fiber.
• $\overline{M}^{(μ)}_{g,n,k,\beta}(X_i, D)$: Relative stable maps to $(X_i, D)$ with $k$ relative marked points of contact order vector $μ$.

These carry virtual fundamental classes with expected dimensions

$$\dim_\mathrm{vir}\overline{M}^{(μ)}_{0,n,k,\beta_i}(X_i, D) = (\dim X_i - 3) + \langle \beta_i, c_1(T_{X_i}) - [D] \rangle + n + k.$$

Evaluation maps target $X_i$ (for absolute markings) and $D$ (for relative markings), enabling descendant and cohomological insertion.

2. Statement of the Degeneration Formula

Jun Li’s degeneration formula for Gromov–Witten invariants in genus zero expresses invariants of the general fiber as a sum over combinatorial data: $$GW^X_{0,m,\beta}(\gamma_1, \dots, \gamma_m) = \sum_{\substack{\beta_1+\beta_2=\beta \ \mu}} \frac{1}{|\mathrm{Aut}(\mu)|} \sum_{I} GW_{0,n_1,k,\beta_1,\mu}^{(X_1, D)}(\gamma_1|_{X_1}, \dots; \delta_{i_1}, \dots, \delta_{i_k}) \; GW_{0,n_2,k,\beta_2,\mu^\vee}^{(X_2, D)}(\gamma_{n_1+1}|_{X_2}, \dots; \delta_{i_1}^\vee, \dots, \delta_{i_k}^\vee).$$ Here, insertions $\gamma_i$ are distributed between $X_1$ and $X_2$, $\mu$ is a contact order partition summing to the intersection number, and the $\delta_{i_j}$ range over a dual basis of $H^*(D)$. The automorphism factor $|\mathrm{Aut}(\mu)|$ corrects for symmetries in the contact data. Each term in the sum is a product of relative invariants paired by duality of insertions at $D$ (Gubarevich, 21 Dec 2025).

The generalization to higher genus and to virtual cycle level replaces classes and invariants with virtual fundamental cycles and Gysin pullbacks along gluing diagonals (Kim et al., 2018).

3. Gluing Morphisms, Obstruction Theory, and Virtual Cycles

The degeneration formula depends crucially on the construction of gluing morphisms between moduli spaces and the compatibility of perfect obstruction theories:

• Relative maps to $(X_i, D)$ and combinatorial splittings (bipartite graphs, contact orders) match along evaluation over $D$.
• The gluing is realized by a fibered product over $D^k$, with combinatorial weights from the contact orders and the structure of the automorphism group.
• The obstruction theory for the total (absolute) moduli stack decomposes into those for the relative moduli, corrected by excess terms pulled back from the diagonal, enabling the application of the virtual pullback and pushforward machinery (Kim et al., 2018).

Logarithmic geometry provides an alternative, where stable log maps and their tropicalizations encode the same combinatorics intrinsically, and gluing is formalized via pushouts of log structures, eliminating the need for expanded targets (Kim et al., 2018).

4. Orbifold and Logarithmic Extensions

In the orbifold setting, Jun Li’s formula is generalized to orbifold targets $G$ and orbifold divisors $Z\subset G$ (Chen et al., 2011). The notation passes to inertia stacks, with moduli like $\mathcal{M}_{g,(g),A,(h),T_k}(G, Z)$ parameterizing stable orbifold maps with specified monodromy and fractional contact orders. The orbifold degeneration formula becomes

$$\left\langle\tau_{d_1}(\alpha_1) \dots \tau_{d_m}(\alpha_m) \right\rangle_{g,A}^{G_t} = \sum_{A^++A^-=A,\,(h),\,T_k} \frac{\prod_{j=1}^k \ell_j}{|\mathrm{Aut}(h, T_k)|} \left\langle\cdots \right\rangle_{g^+, A^+}^{(G^+, Z)} S((h), T_k) \left\langle\cdots \right\rangle_{g^-, A^-}^{(G^-, Z)},$$

where twisted sectors, ages, and orbifold intersection forms enter essentially. The formula specializes to the classical situation when the orbifold structures are trivial and the contact orders are integral.

Logarithmic and analytic frameworks capture the essence of the combinatorial splitting and gluing directly on the central fiber, sometimes using tropical and toric methods for gluing parameters, as in (Tehrani, 2017).

5. Examples, Applications, and Special Cases

A key application of Jun Li's degeneration formula is the computation of enumerative invariants in cases where direct computation is inaccessible. In the context of the genus zero Gromov–Witten theory of even-dimensional complete intersections of two quadrics, the formula was pivotal in establishing the vanishing of a previously inaccessible correlator (Gubarevich, 21 Dec 2025). By specializing the formula to an explicit degeneration—constructing a family where the general fiber is the complete intersection and the special fiber is a union $X_1 \cup_D X_2$—the authors assigned insertions to the two components and, by virtual dimension counting, showed that all contributions vanish.

In orbifold Gromov–Witten theory, the extension accommodates fractional contact orders and twisted sectors, enabling formulas for quotients such as $G = [M/\mathbb{Z}_r]$, where the divisor is $\mathbb{Z}_r$–invariant (Chen et al., 2011, Abramovich et al., 2011). The log and analytic approaches were demonstrated on degenerations with more complex singularities, such as pencils of cubic surfaces and simple normal crossings (Kim et al., 2018, Tehrani, 2017).

6. Technical Refinements and Comparison of Approaches

The derived categories, perfect obstruction theories, and virtual classes in Jun Li’s original algebro-geometric construction (Gubarevich, 21 Dec 2025, Kim et al., 2018) depend on explicit expansions of the target varieties. Logarithmic approaches shift the structural burden to the log structures on the central fiber, encoding combinatorial splittings in facets and cones, with gluing realized as pushouts in the log category. This eliminates the expanded degeneration step and allows more direct combinatorial control (Kim et al., 2018). Analytic and symplectic methods, while closely parallel, achieve similar ends with Gromov compactness and analytic deformation–obstruction theory (Tehrani, 2017).

A central difference in the orbifold context is the accounting for “ghost automorphisms” and the role of root stacks, which replace the pre-deformability condition in Li’s theory with transversality to the twisted locus, streamlining the obstruction-theoretic foundation and making the gerbe structure canonical for automorphism counting (Abramovich et al., 2011).

7. Impact and Continuing Developments

Jun Li’s degeneration formula and its descendants represent a unifying paradigm in modern enumerative geometry, bridging algebraic, symplectic, and analytic geometry, and enabling inductive computations for varieties with complicated degenerations. Its log and orbifold extensions have drastically enhanced applicability, allowing direct handling of singular targets and stacks. The formula underlies the structure of Gromov–Witten/Donaldson–Thomas/Kontsevich–Soibelman wall-crossing frameworks, tropical approaches, and mirror symmetry predictions for degenerating Calabi–Yau and Fano geometries (Chen et al., 2011, Gubarevich, 21 Dec 2025, Kim et al., 2018, Abramovich et al., 2011, Tehrani, 2017). Further directions include refinements for deeper normal crossing degenerations, punctured invariants, and full logarithmic Donaldson–Thomas theory.