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I am certain that the following result holds, but was not able to find a reference. Do you know one? Or maybe you can give a short proof?

Statement. Let $(M^4,\omega)$ be a compact symlectic manifold and let $J_t$ be a smooth family of compatible almost complex structures, $t\in(-1,1)$. Suppose that $S_0$ is a smooth $J_0$-holomorphic sphere in $(M^4,J_0)$ with zero self-intersection.

Consider now $M^4\times (-1,1)$, introduce $J_t$ on each fibre $M^4\times t$, and take in it the surface $S^2_0\times 0$. Then there is a neighbourhood $U$ of $S^2_0\times 0$ such that for any $(x,t)\in U$ there is a unique $J_t$-holomorphic sphere passing through $(x,t)$ contained in $U$. In other words $J_t$-holomorphic spheres produce on $U$ a structure of a smooth $S^2$-fibration (over a $3$-ball).

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    $\begingroup$ Why wouldn't this follow immediately from "automatic transversality"? $\endgroup$
    – Chris Gerig
    Commented Oct 19, 2017 at 1:40
  • $\begingroup$ Dear Chris, many thanks for your comment! Do you think you can give me some relatively pedagogical (or relatively precise) reference from which it will be clear that the statement indeed follows from "automtatic transversality". Or maybe develop your comment into a slightly longer answer? My knowledge in PDEs is poor... I guess I would like to have something line "automatic transverality with parameters, because usually it is stated for $J$ fixed $\endgroup$
    – aglearner
    Commented Oct 19, 2017 at 9:57
  • $\begingroup$ I don't know if it does follow, but it's what popped into my mind (a reference is given in another MO question of yours: J on $CP^2$ that are not tamed). It says that when I perturb $J$ to $J'$ there is a $J'$-holomorphic sphere isotopic to $S^2_0$ (call it still $S^2_0$). So we get the 1-parameter family of $J$-holomorphic spheres in $M\times(-\epsilon,\epsilon)$. I think if $S^2_0$ has a neighborhood $U_0$ in $M$ for which there are unique $J$-holomorphic spheres through each point of the neighborhood, we would get the same statement in the neighborhood $U_0\times(-\epsilon,\epsilon)$. $\endgroup$
    – Chris Gerig
    Commented Oct 19, 2017 at 10:03
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    $\begingroup$ When you vary $J$ the Fredholm index of the deformation operator of the curve doesn't change (functional analysis), so transversality still holds (this is "generic transversality" in McDuff-Salamon's book on $J$-holomorphic curves). $\endgroup$
    – Chris Gerig
    Commented Oct 19, 2017 at 11:25
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    $\begingroup$ Check out the article link.springer.com/article/10.1007/BF02921708 by Hofer-Lizan-Sikorav on automatic transversality, and in particular the proposition on the bottom of p. 155. This tells you that the infinitessimal variations are of the correct dimension, and also non-vanishing, thus giving rise to the sought smooth foliation. $\endgroup$
    – Nikolaki
    Commented Nov 2, 2017 at 22:00

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