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. Thus
$$
S^2=-2,\qquad F^2=0,\qquad S\mathbin{\cdot}F=1.
$$
The 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 $S\mathbin{\cdot}\Sigma\geq0$. Homologically, however,
$$
S\mathbin{\cdot}\Sigma=S\mathbin{\cdot}(S+F)=-2+1=-1,
$$
a contradiction. Hence this example cannot admit such a simultaneous compatible structure.

Solution

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

{c}
{title2=$C_1\mathbin{\cdot}C_2\geq0$}

Distinct simple $J$-holomorphic curves in an almost complex four-manifold have positive local intersection multiplicities. Consequently their homological intersection number is nonnegative whenever they intersect in finitely many points.


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 Incoming links: Positive intersection of J-holomorphic curves