Representation by Complex Surfaces Consider the topological manifold $M=\mathbb CP^2\#\overline{CP^2}$.  This has $H_2(M,\mathbb Z)=\mathbb Z H+\mathbb Z E$ where $H^2=1$ and $E^2=-1$ are the standard generators of the homology of $\mathbb CP^2$ and $\overline{\mathbb CP^2}$ respectively, each with self-intersection as given.
The class $5H-E$ is representable by a complex curve with respect to the standard complex structure on $M$ induced by blow-up from $\mathbb CP^2$.  This curve has genus 6 by adjunction.
Now consider the class $H-5E$.  As $M$ admits an orientation-reversing diffeomorphism, this class is also smoothly represented by an embedded surface of genus 6.
Does $M$ admit a complex structure, inducing the opposite orientation of the standard complex structure, in which this class has a complex representative?  If so, what is the corresponding first Chern class of this structure?
 A: The answer is yes. It is a somewhat tautological construction.
Suppose $f : M \to M$ is a diffeomorphism and $\mathcal{A} = \{(U_i, \varphi_i)\}_{i\in I}$ is a complex atlas. Then $\mathcal{A}' = \{(f^{-1}(U_i), \varphi_i\circ f)\}_{i \in I}$ is also a complex atlas on $M$ and $f : (M, \mathcal{A}') \to (M, \mathcal{A})$ is a biholomorphism, so $c_1(M, \mathcal{A}') = f^*c_1(M, \mathcal{A})$. Moreover, if $i : \Sigma \to (M, \mathcal{A})$ is an embedding of a complex submanifold of $(M, \mathcal{A})$, then $f^{-1}\circ i : \Sigma \to (M, \mathcal{A}')$ is also an embedding of a complex submanifold of $(M, \mathcal{A}')$.
There is an orientation-reversing diffeomorphism $f : \mathbb{CP}^2\#\overline{\mathbb{CP}^2} \to \mathbb{CP}^2\#\overline{\mathbb{CP}^2}$ which 'switches summands', so $f_*$ swaps $H$ and $E$. If $\mathcal{A}$ is the complex atlas from the standard complex structure on $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$ (i.e. the blowup of $\mathbb{CP}^2$ at a point), then $c_1(\mathbb{CP}^2\#\overline{\mathbb{CP}^2}, \mathcal{A})$ is the Poincaré dual of $H - E$. On the other hand, $\mathcal{A}'$ defines a complex structure on $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$ which induces the opposite complex structure, and $c_1(\mathbb{CP}^2\#\overline{\mathbb{CP}^2}, \mathcal{A}')$ is the Poincaré dual of $E - H$.
Suppose $i : \Sigma \to (\mathbb{CP}^2\#\overline{\mathbb{CP}^2}, \mathcal{A})$ is an embedding of a genus $6$ curve with $i_*[\Sigma] = 5H - E$. Then $f^{-1}\circ i : \Sigma \to (\mathbb{CP}^2\#\overline{\mathbb{CP}^2}, \mathcal{A}')$ is an embedding of a genus $6$ curve with $$(f^{-1}\circ i)_*[\Sigma] = (f^{-1})_*\circ i_*[\Sigma] = (f_*)^{-1}(i_*[\Sigma]) = (f_*)^{-1}(5H - E) = -5E + H = H - 5E$$ as desired.
