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Reference: McDuff-Salamon's book J-holomorphic curves and Symplectic Topology, second edition.

In section 6.7, the book introduces the moduli space of $\{J_z\}$-holomorphic curves (see page 184 for example), where $\{ J_z \}_{z\in \Sigma}$ is a smooth family of almost complex structure parametrized by the point $z$ of the domain $\Sigma$. What could be advantages of this compared to the ordinary moduli space for an almost complex structure $J$ independent of $z$?

In particular, when this book discusses the gluing theorem in chapter 10, we also consider parametrized almost complex structures. How does such a slightly general gluing theorem apply more widely than the gluing theorem only for a fixed almost complex structure $J$? I am just wondering if it is really necessary to consider the most general cases at the cost of simplicity.

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    $\begingroup$ To show that the floer homology is independent of the chosen acs, one needs to consider families of such. $\endgroup$
    – Thomas Rot
    Mar 20, 2017 at 6:14
  • $\begingroup$ And when dealing with group actions (an equivariant scenario), you'll want domain-dependent $J$ to help with transversality issues. $\endgroup$ Mar 20, 2017 at 7:16
  • $\begingroup$ In projecteuclid.org/euclid.jsg/1210083200, these domain dependent almost complex structures have been applied to obtain enough transversality to define a version of genus zero Gromov-Witten invariants. Perhaps this could be more easily done by considering inhomogeneous perturbations of the equation as is done in work of Ruan-Tian, FOOO, Pardon etc. In any case, to obtain transversality for GW theory, one seems to need to consider some generalization of the usual pseudoholomorphic curve equation for surface-independent J. $\endgroup$ Mar 20, 2017 at 19:41

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Sorry if I repeat answers already given. This is an answer I got from Auroux's lectures notes on Fukaya category. In defining the Lagrangian intersection Floer homology, an immediate issue is the transversality of the intersection. But that is not all: even if you have transverse intersection, you need transversality of the moduli space $\mathcal{M}(p,q;\beta,J)$ of $J$-holomorphic strips between $p$ and $q$ to ensure that it is a manifold.

Here, transversality means that at each $u \in \mathcal{M}(p,q;\beta,J)$, the linearised operator of $\bar{\partial}_J$ is surjective. This property does not hold in general for a fixed almost-complex structure $J$. However, it does hold for a generic family of almost-complex structure $(J_z)_{z \in \Sigma}$. Depending on what you look at, you may need only a dependence on one real parameter etc.

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