Suppose that $(M,\omega)$ is a symplectic manifold, and $N \subset M$ is a closed submanifold of $M$. If $J_N: TM|_N \rightarrow TM|_N$ is an $\omega$-compatible almost complex structure, defined on $N$, is it always possible to extend $J_N$ to a globally defined $\omega$-compatible almost complex structure?
I feel like this should be the case, since for any vector space $(V,\sigma)$, the space of $\sigma$-compatible almost complex structures is contractible, and so the bundle over $M$ whose fiber at $p$ is the space of $\omega_p$-compatible complex structures of $T_pM$ has contractible fibers, and so every partially-defined section on nice enough closed sets should extend by general homotopy theoretic arguments, but I confess that I'm not comfortable with these sorts of arguments yet, so it would be nice if someone could tell me if something like this works.
Best Answer
It is indeed the case that a compatible/tame almost complex structure defined on an appropriate subspace of a symplectic manifold extends to a compatible/tame almost complex structure over the whole manifold, essentially for the reason you contemplate. (An instance of this result is the object of Exercise 2.2.19 in Chris Wendl's Lectures on Holomorphic Curves in Symplectic and Contact Geometry, when the subspace is some open heighborhood of a closed subset of the symplectic manifold.)
To fix ideas, let's consider the situation for compatible structures; the case of tame structures follows the same reasoning. Given a symplectic manifold $(M, \omega)$, consider the fiber bundle $\pi : E := J(M, \omega) \to M$ whose fiber $E_m$ consists in the complex structures on $T_m M$ which are compatible with $\omega_m$. Using Darboux's theorem, one can show that this is indeed a fiber bundle in the sense given e.g. on pages 376-377 of Allen Hatcher's Algebraic Topology. Hence Proposition 4.48 in Hatcher proves that $\pi$ has the homotopy lifting property with respect to any $CW$-pair $(X, A)$ (see p. 376 for a definition), in particular when $X = M$ and $A$ is some $CW$-subspace of $M$, e.g. when $A = N$ is an embedded closed submanifold.
Now, a compatible almost complex structure on $A$ is a section $J_A$ of $\pi$ over $A$. Moreover, Theorem 2.2.8 in Wendl shows that the space of compatible almost complex structures on $M$, i.e. of global sections of $\pi$, is nonempty and contractible; In fact, it follows from the proof that this contraction occurs fiberwise. This implies that there exist a global compatible almost complex structure $J'$ on $M$ and a fiber-preserving homotopy from $J_A$ to $J'|_A$ over $A$ (and thus a fiber-preserving homotopy from $J'|_A$ to $J_A$). Applying the homotopy lifting property of $\pi$ with respect to $(M, A)$ yields a global section $J$ of $\pi$ such that $J|_A = J_A$.