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Summarising the discussion above and Daniel Ruberman's helpful clarifications below.

Any symplectic structure on $\mathbb{C}P^2$ which admits an embedded sphere 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.)

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.)

Summarising the discussion above and Daniel Ruberman's helpful clarifications below.

Any symplectic structure on $\mathbb{C}P^2$ which admits an embedded sphere 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 automatic transversality, 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.

Summarising the discussion above and Daniel Ruberman's helpful clarifications below.

Any symplectic structure on $\mathbb{C}P^2$ which admits an embedded sphere 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.)

Summarising the discussion above: Any symplectic structure on $\mathbb{C}P^2$ which admits an embedded sphere of self-intersection $+1$ is standard by a theorem of McDuff. Gromov's original paper shows that such a sphere moreover must be smoothly unknotted, i.e.~isotopic to the standard line at infinity. Thus: If we take $S \subset \mathbb{C}P^2$ a smooth sphere whose complement is not is simply connected, no tamed almost complex structure exists for which $S$ is pseudoholomorphic.

Summarising the discussion above and Daniel Ruberman's helpful clarifications below.

Any symplectic structure on $\mathbb{C}P^2$ which admits an embedded sphere 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 automatic transversality, 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.