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Let $J$ be an almost complex structure on $\mathbb{CP}^3$ whose Chern classes are the same as those of the standard complex structure. Let $A$ be the generator of $H_2(\mathbb{CP}^3;\mathbb{Z})$, i.e. the homology class of lines.

I would like to show that the moduli space of $J$-holomorphic spheres representing the homology class $A$ is smooth.

My guess is that every $J$-holomorphic $A$-sphere $u$ is immersed and has pullback bundle $$u^{\ast}T\mathbb{CP}^3=\mathcal{O}(2)\oplus\mathcal{O}(1)\oplus\mathcal{O}(1).$$ So if I could show that each summand is invariant under $D_u$, I could apply Lemma 3.3.2 from McDuff and Salamon to conclude that $D_u$ is surjective (i.e. the moduli space is smooth). The first summand is invariant by the definition of $D_u$. What about the other two summands?

Is this reasoning correct?

Note that if $J$ is the standard complex structure, then the moduli space is $\mathrm{Gr}_2(\mathbb{C}^4)$ (equivalently, the Klein quadric $Q_4\subset\mathbb{CP}^5$), as shown in the answer to this question.

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