Simple J-holomorphic curve

A J-holomorphic curve is simple if it does not factor through a holomorphic branched covering of degree greater than one. For a nonconstant curve this is equivalent to having a point where its derivative is nonzero and its image has exactly one preimage.

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.

The singular-curve adjunction formula in a symplectic manifold of real dimension four gives $c_1(A)=A^2+2-2\delta$ for a simple J-holomorphic sphere, with a nonnegative singularity count $\delta$. Hence the displayed inequality holds. This is the version needed when the sphere need not be embedded.