The connect sum $X:=CP^2\# CP^2 \# CP^2$ supposedly supports almost-complex structures, i.e. endomorphisms $J$ of the tangent bundle such that $J^2=-id$. The existence of these almost-complex structures has been discussed previously on MO, e.g. Peter Teichner's answer to A question on classification of almost complex structures on $4$-manifolds.

I would like to have a concrete description of at least one of these almost-complex structures on $3 \#CP^2$. E.g. Concrete enough to have a chance of constructing lots of sections of the `subbundle' $C_J$ of $\wedge^2TX$, where $(C_J)_x$ consists of all $J_x$-invariant two-planes in $T_xX$ for $x\in X$.

Can anybody direct me to a reference or construction which explicitly yields some almost-complex structure on $X$ ?


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