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Take the Hirzebruch surface $X={\mathbb{F}}_2$ with its standard toric Kähler structure. Let $C_1=S$ be its negative section of a Hirzebruch surface, and let $C_2=F$ represent the fiber class of a Hirzebruch surface and meet $S$ once. ThusThe preceding smoothing produces an embedded symplectic $\Sigma$ in class $S+F$.

Suppose a compatible almost complex structure made all three surfaces almost complex. Since $S$ and $\Sigma$ are distinct, Positive intersection of J-holomorphic curves would imply $S\cdot \Sigma \ge 0$. Homologically, however,a contradiction. Hence this example cannot admit such a simultaneous compatible structure.

On ${\mathbb{F}}_2$, let $S$ be the negative section of a Hirzebruch surface, so $S^2=-2$, and let $F$ be the fiber class of a Hirzebruch surface, with $F^2=0$ and $S\cdot F=1$. The line bundle $\mathcal{O}(S+3F)$ is very ample. Indeed, the toric ampleness criterion for $aS+bF$ isand on a smooth complete toric variety every ample line bundle is very ample. These inequalities hold for $(a,b)=(1,3)$.

Explicitly, after choosing the standard lattice coordinates for ${\mathbb{F}}_2$, its lattice polytope of a toric divisor isThe monomials indexed by the lattice points of this polygon separate torus orbits and tangent directions, so their Kodaira map is a closed embedding. This directly verifies that the resolved proper toric surface is projective.

For the Hirzebruch surface ${\mathbb{F}}_n$, the negative section of a Hirzebruch surface $S$ and a fiber class of a Hirzebruch surface $F$ freely generate $\mathrm{Pic}({\mathbb{F}}_n)$ and satisfy