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  "body": "\n## Summary\n\nA 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.\nOn a symplectic manifold with a compatible almost complex structure, the moduli spaces of such curves carry topological information.\nGromov introduced them in 1985, and they are the basis of Gromov–Witten invariants, [[Floer homology|NCKFPAGHXF]] and [[Fukaya categories|4HS60JABC5]].\n\n## Explanations\n\n### L1 human\n\nA holomorphic function of one complex variable bends the plane in a way that keeps angles.\nA 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.\nOn a symplectic manifold these curves behave like soap films: they minimize area in their class.\nFor example, in $\\C^2$ with the usual complex structure, the map $z \\mapsto (z, z^2)$ is a pseudoholomorphic curve.\n",
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