Real Pseudoholomorphic Curves

• Real pseudoholomorphic curves are J-holomorphic maps with a compatible real structure, extending classical real algebraic curves.
• They reveal key topological invariants through complex orientation and separating properties that distinguish symplectic from algebraic realizability.
• Their study employs advanced analytic techniques, including elliptic regularity and local parametrizations, with applications in contact geometry and real embedded contact homology.

A real pseudoholomorphic curve is a $J$-holomorphic curve equipped with a real structure, typically formulated as the fixed-point set of an anti-involution on a symplectic manifold or complex analytic space. Real pseudoholomorphic curves generalize real algebraic curves through the flexibility of varying almost complex structures while preserving anti-invariance under a symmetry, revealing subtle topological and enumerative features unattainable or obstructed in algebraic geometry. A central theme is the contrast between algebraic realizability and symplectic realizability, especially with regards to orientation, isotopy, and moduli.

1. Foundational Definitions and Structures

Let $(X, \omega)$ be a compact symplectic $4$-manifold, $c: X \to X$ a smooth involution such that $c^*\omega = -\omega$, and $J$ an $\omega$-tame almost complex structure satisfying $c^*J = -Jc^*$. A real pseudoholomorphic curve is a smooth, $c$-anti-invariant $J$-holomorphic submanifold $A \subset X$ of real dimension $2$. The set $RA = A \cap \text{Fix}(c)$, i.e., the real locus, is a smooth one-dimensional submanifold in $RP^2$ if $X = CP^2$ equipped with standard conjugation and Fubini–Study symplectic form (Orevkov, 2020). The degree $m$ of $A$ is given by $[A] = m[CP^1]$ in $H_2(CP^2)$.

If $(\Sigma, J_\Sigma)$ is an oriented Riemann surface and $\phi: (\Sigma, J_\Sigma) \to (X, J)$ solves $d\phi \circ J_\Sigma = J \circ d\phi$, then $\phi(\Sigma)$ is a (pseudo)holomorphic curve. The curve is real pseudoholomorphic if there exists anti-holomorphic involutions on both domain and target such that $\phi$ is equivariant (Brugalle, 2014). The fixed locus $\phi(\mathbb{R}\Sigma) \subset \mathbb{R}X$ forms the "real part" of the curve.

2. Separating Curves and Complex Orientation Invariants

A separating (Type I) real curve $A \subset CP^2$ is non-singular, such that $A \setminus RA$ consists of two connected components permuted by $c$. Equivalently, $RA$ is null-homologous in $A$, and $A \setminus RA$ splits into two complex-conjugate halves (Orevkov, 2020).

The complex orientation of the real locus $RA$ is induced as follows: each of the two components of $A \setminus RA$ inherits the natural complex orientation as a $J$-holomorphic curve, and their boundaries endow $RA$ with orientations. An oval $O$ (compact component of $RA$) may be positive or negative depending on the induced boundary orientation's homology class. The enumeration of positive/negative even and odd ovals ($A_p^+, A_p^-, A_n^+, A_n^-$) provides key invariants.

In pseudoholomorphic curve theory, these orientation data refine the understanding of the real locus beyond classical algebraic constraints, leading to powerful obstruction criteria and topological classifications.

3. Complex Orientation Inequalities and Algebraic Realizability

For a non-singular separating real algebraic curve $A \subset RP^2$ of odd degree $m = 2k+1$, Theorem 1.1 establishes two orientation inequalities:

• Left: $A_p^+ + A_p^- + 1 \geq l - k(k-1)$
• Right: $A_n^+ + A_n^- - 1 \geq k(k+1) - l$

with $l = r-1$ the number of ovals, $r$ the number of components of $RA$, $J$ the pseudo-line, and further restatements in terms of genus and discrete invariants (Orevkov, 2020).

Violation of these inequalities in a separating real pseudoholomorphic curve implies its complex scheme (oriented isotopy type of its real locus) is algebraically unrealizable: no genuine real algebraic curve of equal degree can realize this topological data. This is notably exhibited by constructions in degree $9$ and, in general, for degrees $m \equiv 9 \pmod{12}$ via the tripling–cubic perturbation. These symplectic constructions exploit topological and almost-complex freedom unattainable in strictly algebraic settings.

4. Analytic Framework: Local Parametrizations and Regularity

Pseudoholomorphic curves are analytically governed by nonlinear Cauchy–Riemann equations. For a map $u: \Omega_1 \to \Omega_2$ subject to $\partial u / \partial \bar{z} = E(z, u(z))$ almost everywhere, elliptic regularity ensures measurably weak solutions immediately enjoy higher regularity, with sharp Hölder exponent $\alpha = 1 - 2/p$ for $P \in L^p_\text{loc}$, and real-analyticity if $E$ is analytic (Coffman et al., 2014).

In normal coordinates near an embedded $J$-holomorphic curve, one reduces to block structures and separates the analytic system into decoupled equations, enabling explicit local parametrizations (cf. Lemma 5.1). Such normal forms clarify the analytic flexibility of pseudoholomorphic curves in the almost-complex category.

5. Symplectic and Topological Classification: Simple Harnack Curves

The topological classification of real pseudoholomorphic curves finds full generalization in the context of simple Harnack curves. If $C$ is a maximal real curve of genus $g$, and there exists a distinguished oval and three arcs capturing all intersections with fixed real lines, the combinatorial intersection data $(d, g, s)$ fully determines the topological type up to homeomorphism, independent of the almost-complex structure $J$ (Brugalle, 2014).

Mikhalkin’s theorem, originally for algebraic curves, extends verbatim to the pseudoholomorphic setting: real pseudoholomorphic simple Harnack curves exhibit only solitary node singularities and admit a topological classification in $(RP^2, \mathbb{R}\phi(C) \cup \cup_{i=0}^2 RL_i)$ by degree, genus, and intersection data. This highlights the symplectic and topological core of the classification, disconnected from strict algebraic constraints.

6. Obstructions, Index Theory, and Contact Geometry

In dimension 3, real pseudoholomorphic curves arise in contact geometry equipped with an anti-contact involution. Such curves contribute to real embedded contact homology (ECH) and cylindrical contact homology. The iteration formulae for brake orbits (symmetric Reeb trajectories) establish computes for half-period indices and Conley–Zehnder indices, distinguishing orbits by elliptic and hyperbolic (Type I/II) behaviors (Zhou, 2020). The Real ECH lemma and index inequalities bound the Fredholm index of moduli spaces, which is central for compactness and transversality in holomorphic curve counts, and guarantee uniqueness of partitions for symmetry-related chain complexes.

7. Generalizations, Open Problems, and Future Directions

The analysis of real pseudoholomorphic curves extends to higher degrees, other rational or ruled symplectic surfaces, and higher dimensions. Investigations concern the determination of the full separating semigroup, potential extensions of orientation inequalities to broader classes, and the construction of symplectic topological obstructions to algebraic realizability—especially through operations such as Luttinger surgery. Classifications remain open for degrees $8$ and $10$, and future work will examine the generalization of real curve invariants and moduli in relative and non-integrable settings (Orevkov, 2020).

The study of real pseudoholomorphic curves augments classical real algebraic geometry, providing new pathways for the understanding of symplectic topology, curve enumeration, and topological classification. Prominent avenues involve the application to toric surfaces, integrable systems, spectral curves in physics, and the search for symplectic invariants beyond the reach of algebraic methodologies.