---
tag: N469E47V75
kind: concept
name: Pseudoholomorphic curve
msc: [32Q65, 53D35, 53D45]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudoholomorphic_curve
same_as:
  wikidata: Q2115749
  wikipedia: Pseudoholomorphic curve
---

## Summary

A pseudoholomorphic curve, or $J$-holomorphic curve, is a map from a [[Riemann surface|5K0N8QMV6Q]] into an [[almost complex manifold|9SYWRGVTRS]] whose differential is complex linear.
On a symplectic manifold with a compatible almost complex structure, the moduli spaces of such curves carry topological information.
Gromov introduced them in 1985, and they are the basis of Gromov–Witten invariants, [[Floer homology|NCKFPAGHXF]] and [[Fukaya categories|4HS60JABC5]].

## Explanations

### L1 human

A holomorphic function of one complex variable bends the plane in a way that keeps angles.
A pseudoholomorphic curve is the same idea for maps of a surface into a bigger space that only has a pointwise notion of multiplication by $i$ on tangent vectors.
On a symplectic manifold these curves behave like soap films: they minimize area in their class.
For example, in $\C^2$ with the usual complex structure, the map $z \mapsto (z, z^2)$ is a pseudoholomorphic curve.