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For a regular nonconstant J-holomorphic sphere in a $2n$-dimensional target, the parametrized Fredholm index is $2n+2c_1(A)$. Removing the six-dimensional Möbius transformation reparametrization group gives the displayed local moduli dimension. The quotient is a manifold for a simple J-holomorphic sphere and may be an orbifold for a multiple cover with finite stabilizer. In dimension four the formula is $2c_1(A)-2$.

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.

For a compact symplectic manifold, smooth convergence of tame almost complex structures and a uniform area bound give a subsequence of J-holomorphic spheres converging, after reparametrization, to a finite tree of J-holomorphic spheres, with convergence smooth away from finitely many bubbling points. The total homology class and symplectic area are preserved. Constant components may be added for stability. In a spherical class which cannot split into two positive-area spherical classes, there is only one nonconstant component.

If every nonconstant J-holomorphic sphere under consideration is regular, no such sphere in a four-dimensional target has class $A$ with $A^2\le -2$. For a simple J-holomorphic sphere, the adjunction inequality for a simple J-holomorphic sphere makes the predicted moduli dimension negative. A degree-$m$ multiple cover of class $B$ has $B^2\le -1$ and $c_1(B)\le 1$, so its predicted real dimension is at most $2m-2$. The rational covering maps of the projective line already form a family of dimension $4m-4>2m-2$, contradicting regularity. The full regularity hypothesis matters: multiple covers of exceptional minus-one spheres violate the conclusion under a simple-only regularity assumption.

A degree-$m$ rational map of the complex projective line is represented by two degree-at-most-$m$ polynomials without a common root, modulo common scalar multiplication. This is an open set of ${\mathbb{CP}}^{2m+1}$. Quotienting by domain Möbius transformations subtracts three complex dimensions, giving real dimension $4m-4$ for degree $m\ge 2$. These are the local covering parameters of a multiple-covered J-holomorphic sphere.

A J-holomorphic curve on a punctured disc with finite energy and image contained in a compact target extends smoothly across the puncture when the target has a smooth tame almost complex structure and compatible bounded geometry. Local monotonicity controls escape from small target balls, and the elliptic small-energy estimate followed by elliptic regularity gives the extension. This is the step turning a finite-energy plane obtained by rescaling into a J-holomorphic sphere.

A simple J-holomorphic sphere is a J-holomorphic sphere that is a simple J-holomorphic curve. Its automorphism group as a parametrized map is trivial, because any automorphism fixing the map fixes its locally injective points.