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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.

An almost complex structure $J$ is compatible with $\omega$ if $J^2=-I$, $\omega (Ju,Jv)=\omega (u,v)$, and $g_J(u,v)=\omega (u,Jv)$ is a positive-definite inner product. To construct one, choose a Riemannian metric $h$ and define $A$ by $h(Au,v)=\omega (u,v)$. The map $A$ is invertible and skew-adjoint, so $-A^2$ is positive definite. Its positive square root $P=(-A^2{)}^{1/2}$ commutes with $A$, andThe same identities show that $J$ preserves $\omega$. The positive square root depends smoothly on $A$, giving a smooth compatible almost complex structure.

For a compatible $J$, applying this metric construction of a compatible almost complex structure to $g_J$ returns $J$. Given $J_0,J_1$, apply it to $(1-t)g_{J_0}+t{g}_{J_1}$ to obtain a path joining them. In fact interpolation to any fixed auxiliary metric gives a contraction. Thus the space is nonempty and connected, and even contractible, by contractibility of compatible almost complex structures.

Regularity of a J-holomorphic curve $u$ means that its linearized Cauchy–Riemann operator $D_u$ is surjective. Differentiating in coordinates givesup to the harmless factor $1/2$ in the convention for ${\bar{\partial }}_J$. For constant $u$, the last term vanishes, $J(u)$ is constant, and the pulled-back tangent bundle is ${\mathbb{CP}}^1\times T_{u}M$. Identifying $T_{u}M$ with ${\mathbb{C}}^n$, this is a direct sum of $n$ copies of the Dolbeault operator on the trivial line bundle. Its cokernel is $H^{0,1}({\mathbb{CP}}^1,\mathcal{O})=0$, by Dolbeault cohomology of the projective line. Hence constant maps are regular for every compatible $J$.

The Energy identity for a J-holomorphic curve says that the energy is ${\int }_{{\mathbb{CP}}^1}{u}^{\ast }\omega$. If the homology class is zero, this integral is zero, forcing $du=0$. Thus every zero-class sphere is constant and regular.

In real dimension four, the Fredholm index of the parametrized sphere operator is $4+2c_1(A)$. At a regular nonconstant sphere, quotienting by the six-dimensional Möbius transformation group gives a local moduli space, or an orbifold for finite stabilizers, of real dimensionThis is the dimension formula for regular J-holomorphic spheres.

For a simple J-holomorphic sphere in class $A$, the adjunction inequality for a simple J-holomorphic sphere gives $c_1(A)\le A^2+2$. If $A^2\le -2$, the displayed moduli dimension is negative, so such a regular simple J-holomorphic sphere cannot exist. To cover nonsimple maps as well, interpret regularity here as regularity for all nonconstant maps under discussion. A degree-$m$ cover, $m\ge 2$, of a simple J-holomorphic sphere in class $B$ has $A=mB$, with $B^2\le -1$ and $c_1(B)\le 1$. Its local family of rational covering maps of the projective line modulo domain reparametrization has real dimension $4m-4$, while the predicted dimension isIt therefore cannot be regular. This proves there are no regular spheres in a class with square at most minus two, by negative-square exclusion under full sphere regularity. Regularity only for simple curves would not suffice: multiple covers of an exceptional minus-one sphere are the counterexample.