Recall that an almost complex structure $J$ on a manifold $M^{2n}$ is called *tamed* if there exists a symplectic form $\omega$ on $M^{2n}$ such that $\omega(v,Jv)>0$ for any non-zero tangent vector $v$.


**Question.** Is there an example of an almost complex structure on $\mathbb CP^2$ such that any $C^{\infty}$ small perturbation of $J$ is not tamed?


**Added.** It turns out that there exists as well a purely *local* obstruction for any small perturbations of $J$ to be tamed. The precise statement and the answer is here: [Almost complex structures on a 4-ball that are not tamed](https://mathoverflow.net/questions/263500/almost-complex-structures-on-a-4-ball-that-are-not-tamed)



Almost complex structures on $\mathbb CP^2$ that are not tamed
Summarising the discussion above and Daniel Ruberman's helpful clarifications below.


Any symplectic structure on $\mathbb{C}P^2$ is standard by a result due to Gromov and Taubes. By Siebert-Tian every symplectic surface in $\mathbb{C}P^2$ of degree at most 17 is smoothly isotopic to an algebraic surface. In particular, there is a unique smooth isotopy class of such surfaces. 


Now, take $S \subset \mathbb{C}P^2$ a surface of low degree which is not ambient diffeomorphic to an algebraic surface, but which satisfies the adjunction formula ($g=(d-1)(d-2)/2$). Any almost complex structure for which $S$ is pseudoholomorphic can then *not* be tamed. (To see that such an almost complex structure exists in the right homotopy class, we can find a smooth homtopy of $\mathbb{C}P^2$ from the identity to a smooth map (not a diffeomorphisms obviously!) which sends $S$ to an algebraic surface.)


Finally, by the automatic transversality of Hofer-Lizan-Sikorav, for a small perturbation of the almost complex structure we can find a pseudoholomorphic surface isotopic to $S$; these almost complex structures hence do not admit taming forms either. (In order to apply the automatic transversality, we must use the assumptions that $c\_1([S]) \ge 1$ and that $S$ is immersed.)


