= Fibre spheres in a sphere-torus product
{title2=$F=[S^2\times\{q\}]$}
In a product <symplectic manifold> $S^2(a)\times T^{2n-2}$, every <compatible almost complex structure> admits a <J-holomorphic sphere> in class $F$ through each point. Here $S^2(a)$ has area $a$, and the <torus> carries a constant <symplectic form>. All spherical classes are integer multiples of $F$, so a curve in class $F$ cannot split into positive-area nonconstant spheres. For the product <complex structure> the fibre through a point is unique and regular. Its one-point count modulo two is one; generic <transversality> and the compact one-dimensional parameterized <moduli spaces> preserve this count. Approximation and <Gromov compactness for spheres> remove the genericity requirement. This gives the area-$a$ curve used in the proof of the <Gromov non-squeezing theorem>.
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