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Let $(M, \omega)$ be a symplectic 4-manifold and let $A$ and $B$ be symplectic submanifolds on M such that $A \cap B = p \in M$.Under what conditions can I find a $\omega$-compatible almost complex structure $J$ such that both $A$ and $B$ are J-holomorphic.

I understand that one would require certain necessary conditions like $[A] \cdot [B] = 1$ (In order to satisfy positivity on intersections in dimension 4) but is this condition sufficient?

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    $\begingroup$ You need to assume transverse intersection. Then this can be achieved for $\omega$-tame J (see for example Lemma 3.1 of McDuff’s "Immersed spheres in symplectic 4-manifolds") but off the top of my head I’m not sure about subtleties for tame-to-compatible. $\endgroup$
    – Chris Gerig
    Commented Feb 26, 2019 at 2:49
  • $\begingroup$ Thank you. I'll have a look at it. $\endgroup$
    – cr1t1cal
    Commented Feb 26, 2019 at 2:52

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