Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 146 / 2 / b / Solution

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.