Pseudoholomorphic Curve Invariants

• Pseudoholomorphic curve invariants are integer or rational counts obtained by signing J-holomorphic curves in symplectic manifolds, remaining invariant under deformation.
• They employ advanced moduli space constructions and Fredholm theory with transversality methods like Kuranishi atlases to ensure robust, virtual dimension-zero counts.
• These invariants connect symplectic topology to enumerative geometry and gauge theory, aiding in embedding obstructions, singularity analysis, and correspondence theorems.

A pseudoholomorphic curve invariant is an integer- or rational-valued quantity derived from signed counts of $J$-holomorphic curves in a symplectic manifold, subject to homological, topological, or geometric constraints, and invariant under deformation of the background geometric data. These invariants encode deep geometric and enumerative information about the ambient symplectic manifold, singularity behaviour, and, in many situations, functional relations with other curve-counting theories such as Gromov–Witten, Gopakumar–Vafa, Donaldson–Thomas, and Seiberg–Witten invariants. Modern developments have led to the construction of robust classes of pseudoholomorphic curve invariants that incorporate curves with singularities, punctured curves with prescribed asymptotics, and refinements sensitive to wall-crossing and higher genus phenomena.

1. Foundations: Moduli Spaces and Fredholm Framework

The construction of pseudoholomorphic curve invariants begins with the Fredholm theory of $J$-holomorphic maps $u: \Sigma \to (M, \omega)$, for a closed symplectic $2n$-manifold $(M,\omega)$ and compatible almost complex structure $J$. The relevant moduli spaces are:

• $\overline{\mathcal{M}}_{g, k}(M, A)$: stable genus-$g$ degree-$A$ maps with $k$ marked points, modulo automorphisms.
• For curves with singularities (e.g., cusps), one imposes multidirectional tangency constraints $C^m$: $u([0:0:1]) = p$, with contact orders $\operatorname{ord}_{z=0}(u^*D_s) \geq m_s$, $1 \leq s \leq n$ for divisors $D_s$ through $p$ spanning $T_pM$ [(McDuff et al., 2023), Def. 2.1].

For punctured curves, moduli spaces such as $\mathcal{M}_{M \backslash E(a), A}^J(o_k)$ parametrize $J$-holomorphic planes on the complement of a symplectic ellipsoid neighbourhood, asymptotic to Reeb orbits $o_k$ on the boundary $\partial E(a)$, with index determined in terms of $c_1(A)$. The Fredholm index, contact condition codimension, and automorphism behaviour play essential roles in ensuring robust virtual dimension-zero counts [(McDuff et al., 2023) §1, (Gerig et al., 2014)].

The establishment of an appropriate virtual fundamental class or a transversality method (e.g., Kuranishi atlases, normally complex sections (Parker, 2013), or analytic perturbation (Gerig et al., 2014)) ensures that signed counts and associated invariants are independent of choices and deformation-invariant in suitable settings.

2. Classes of Invariants: Singular, Punctured, and Superpotential Types

The modern landscape includes several robust types of pseudoholomorphic curve invariants:

• Cusp-curve counts $N_{M, A} C^m$: Counts of genus-zero $J$-holomorphic maps in $A$ with a prescribed tangency constraint $C^m$ at $p$. For $m = (p,q,1,\dots,1)$, $p+q = c_1(A)$, $\gcd(p,q) = 1$, and $M$ semipositive, Theorem 2.2 of (McDuff et al., 2023) ensures that $N_{M, A} C^{(p,q)} \in \mathbb{Z}$ is well-defined and deformation-invariant.
• Ellipsoidal superpotentials $T_{M, A}^a$: Integer counts of punctured $J$-holomorphic planes in $M \setminus E(a)$, negatively asymptotic to the Reeb orbit $o_{c_1(A)-1}$, under suitable partition and rational-independence conditions on $a$ [(McDuff et al., 2023), Thm 1.7]. These are assembled into generating functions $W_E(x, y) = \sum_{k, l \geq 0} N_{k, l}^E x^k y^l$.

An explicit equivalence is established between these two frameworks: for suitable $a$ and $m$, $T_{M,A}^a = N_{M,A}C^m$ [(McDuff et al., 2023), Thm 3.1]. This provides a bijection between the foundational moduli spaces and identifies their signed counts.

In higher-genus or more general settings, integral curve counts as constructed via normally complex perturbations yield invariants $I_{g,A}$ for general moduli of curves in a compact symplectic manifold (Parker, 2013).

3. Deformation Invariance and Transversality Methods

Ensuring deformation invariance necessitates advanced transversality techniques:

• Normally complex sections: For each Kuranishi chart, the use of a section locally holomorphic transverse to automorphism strata achieves robust integer counts. Real codimension at least 2 prevents bad boundary phenomena (Parker, 2013).
• Taubes-style analytic perturbations: For unbranched multiple covers, the Cauchy–Riemann operator is perturbed by an antilinear bundle map, allowing the use of the Bochner–Weitzenböck formula and analytic Fredholm theory to achieve regularity for all unbranched index-zero covers (Gerig et al., 2014). As a result, genus-zero GW invariants in dimension four are computed by finite signed and weighted sums over embedded and unbranched covered $J$-holomorphic curves.
• Exploded manifolds framework: As in (Parker, 2013), the exploded category handles normal-crossing degenerations and gluing behaviour smoothly, ensuring invariance under deformations and stable behaviour under limits or wall-crossing.

Deformation invariance is established by extending the moduli problem over parameter spaces and verifying that contributions from boundary strata are invisible for dimension-zero moduli (due to codimension considerations).

4. Applications: Embedding Obstructions, Enumerative Geometry, and Correspondence Theorems

Pseudoholomorphic curve invariants have rich implications:

• Symplectic embedding obstructions: If $T_{M,A}^a \neq 0$, any symplectic embedding of $E(c)\times\mathbb{C}^N \to M\times\mathbb{C}^N$ must satisfy $c \leq [\omega_M]\cdot A / A(o_{c_1(A) - 1})$ [(McDuff et al., 2023), Cor. 1.9]. This generalizes classical capacities—existence of suitable curves with prescribed singularities or asymptotics gives sharp obstructions.
• Singular symplectic curves: The existence of a genus-zero rational symplectic curve in $M$ with exactly one $(p,q)$ cusp and suitable nodal data is equivalent to positiveness of $T_{M,A}^a$, tying geometric degeneracy types to invariants [(McDuff et al., 2023), Thm 4.1].
• Algebraic correspondences and uniqueness: On the first Hirzebruch surface $F_1$, a rigid index-zero unicuspidal rational curve of type $(p,q)$ in class $A=d\ell-me$ exists if and only if $A$ is a $(p, q)$-perfect exceptional class; then $N_{F_1, A} C^{(p,q)} = 1$. The infinite family of possible such classes is enumerated by recursive formulas with clear combinatorial structure [(McDuff et al., 2023), Thm 5.4, Cor. 5.6].

These invariants provide a geometric bridge from SFT curve counts and local algebraic geometry to symplectic topology and embedding problems.

5. Broader Context: Connections to Other Invariant Theories

Pseudoholomorphic curve invariants are central to the interface between symplectic topology, algebraic geometry, and gauge theory:

• Gromov–Witten and Reduced Theories: In holomorphic symplectic contexts, reduced Gromov–Witten invariants and their integer transforms (Gopakumar–Vafa type) operate in this framework, often relying on cosection localization and correspondence theorems (Cao et al., 2022).
• Donaldson–Thomas Correspondences: Many invariants admit sheaf-theoretic interpretations, with reduced DT$_4$ classes on the moduli of one-dimensional sheaves mirroring curve count formulas (Cao et al., 2022).
• Near-symplectic and ECH invariants: Taubes’s “SW=Gr” principle extends to near-symplectic 4-manifolds via punctured curve counts in the complement of degeneracy loci, encoding Seiberg–Witten invariants in curve-theoretic terms (1711.02069).
• Spectral Invariants and Dynamics: Max-min energy invariants for pseudoholomorphic curves provide complete dynamical characterizations of contact forms (e.g., Zoll, Besse) in three dimensions, linking curve energies to Reeb orbit properties via spectral gap principles (Fernandes et al., 7 Oct 2025).

These correspondences validate pseudoholomorphic curve counts as integral to the structure of symplectic and enumerative geometry.

6. Examples and Explicit Computations

Explicit enumerative results are established in key examples:

Manifold/Setting	Invariant/Count	Key Property
$\mathbb{P}^2$	$n_{0,1} = 1$, $n_{0,2} = 1$	Standard line and conic counts (Parker, 2013)
First Hirzebruch surface $F_1$	$N_{F_1,A}C^{(p,q)} = 1$	Uniqueness in each perfect exceptional class (McDuff et al., 2023)
Kodaira–Thurston nilmanifold	$GW_{1, [m,m,n,n]}$ as closed formula	First genus-1 non-Kähler family computation (Evans et al., 2012)

These examples both validate universality of the constructions and illuminate subtleties arising from symplectic structure and singularity constraints.

7. Extensions and Active Research Directions

Current research explores extensions along several axes:

• Higher genus and wall-crossing: Potential for generalizing the integral invariants and their correspondence to wall-crossing and higher genus settings, as seen in the context of chambered invariants for real Cauchy–Riemann operators, which track curve counts via moduli of PDE solutions across wall crossings (Doan et al., 28 Oct 2024).
• Exploded and non-Kähler settings: Count invariants in the exploded manifold category, and extension to non-Kähler and near-symplectic manifolds.
• Holomorphic anomaly and modularity: In fourfolds, generating functions for reduced GW invariants are conjectured to be quasi-Jacobi forms, governed by holomorphic anomaly equations (e.g., in the K3$^{[2]}$ case) (Cao et al., 2022).
• Embedded contact homology and SFT: Connections with ECH, SFT, and their spectral invariants continue to be developed, providing deeper links between symplectic and contact topology and pseudoholomorphic curve enumeration.

Central open problems include universality of integrality for higher genus invariants, explicit classification in higher dimensions or singularity types, and the discovery of new geometric applications for these refined invariants.