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An almost complex structure $J$ on $(M,{\omega }_M)$ is a compatible almost complex structure when $J^2=-I$, ${\omega }_M(Ju,Jv)={\omega }_M(u,v)$, andis a positive-definite inner product. To prove existence, choose any Riemannian metric $h$ and define $A$ by ${\omega }_M(u,v)=h(Au,v)$. The metric construction of a compatible almost complex structureis smooth and compatible, so the space is nonempty.

Identify the compact symplectic manifold $N$ with its image under the symplectic embedding. Along $N$ there is a symplectic splittingChoose compatible almost complex structures on both summands and take their direct sum. Its associated metric makes the two summands orthogonal. Extend this metric from the closed submanifold $N$ to all of $M$ using a partition of unity, and apply the metric construction again. Along $N$ it recovers the prescribed direct sum, so the resulting global compatible $J$ satisfies $J(TN)=TN$. This is the relative extension of a compatible almost complex structure.

For the two compact complex curves, use the supplied holomorphic coordinates at their transverse intersection. There $C_1\cup C_2$ is $\{xy=0\}$. Replace it in a small ball by the complex annulus $\{xy=ϵ\}$ and use a cutoff in a surrounding annulus to rejoin the unchanged curves. For sufficiently small nonzero $ϵ$, the result is an embedded symplectic surface $\Sigma$. This local replacement is a bordism between the old and new cycles, soThis is the symplectic smoothing of a positive transverse node.

For a connected embedded symplectic surface $C$ in a symplectic four-manifold,where $K_{\omega }=-c_1(TM,J)$ is the symplectic canonical class for any compatible almost complex structure.

The symplectic canonical class is the negative First Chern class of the tangent bundle for a compatible almost complex structure. The contractibility of compatible almost complex structures makes it independent of that choice. It is the class appearing in the symplectic adjunction formula.

In the maximal-dimension convention, a totally real submanifold of $(M,J)$ has half the real dimension of $M$ and $T_{p}L\cap J(T_{p}L)=0$. Every Lagrangian submanifold is totally real for a compatible almost complex structure, but the converse fails in complex dimension at least two.

Two disjoint symplectic balls of radii $r_1,r_2$ embedding into the unit four-ball must satisfy the displayed inequality. Extend their transported standard complex structures on smaller concentric balls to a compatible almost complex structure on the projective plane. A degree-one J-holomorphic curve through both centres has area $\pi$. The Monotonicity theorem for a J-holomorphic curve bounds its area in each smaller ball below by $\pi {\rho }_i^2$. Disjointness allows these bounds to be added; letting ${\rho }_i$ increase to $r_i$ proves the result.