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  "body": "\n## Definition\n\nLet $(M, J)$ be an [[almost complex manifold|9SYWRGVTRS]], so $J$ is a smooth endomorphism of $TM$ with $J^2 = -\\id$.\nLet $(\\Sigma, j)$ be a [[Riemann surface|5K0N8QMV6Q]], with $j$ its complex structure on $T\\Sigma$.\nA *pseudoholomorphic curve* (or *$J$-holomorphic curve*) is a smooth map $u \\colon \\Sigma \\to M$ such that\n$$du \\circ j = J \\circ du.$$\n\n## Remarks\n\nIn a local coordinate $z = s + it$ on $\\Sigma$, the condition reads $\\partial_s u + J(u) \\, \\partial_t u = 0$, a nonlinear Cauchy–Riemann equation.\n\nExample: let $M = \\C^n$ with $J$ the multiplication by $i$, and $\\Sigma$ an open subset of $\\C$.\nThen $u$ is pseudoholomorphic exactly when each component is a holomorphic function, by the Cauchy–Riemann equations.\nSo $u(z) = (z, z^2)$ is a pseudoholomorphic curve in $\\C^2$.\n\nNon-example: $u(z) = \\bar z$ from $\\C$ to $\\C$.\nWith $z = s + it$ and target coordinate $x + iy$, one has $du(j \\partial_s) = du(\\partial_t) = -\\partial_y$, while $J\\,du(\\partial_s) = J \\partial_x = \\partial_y$.\nSo $du \\circ j = -J \\circ du \\ne J \\circ du$; the map is antiholomorphic.\n\nIf $J$ is tamed by a symplectic form $\\omega$, meaning $\\omega(v, Jv) \u003e 0$ for $v \\ne 0$, then the energy $\\int_\\Sigma u^* \\omega$ of a nonconstant pseudoholomorphic curve is positive.\nWhen $\\Sigma$ is closed and $\\omega$ is closed, this energy depends only on the homology class of $u$.\n",
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