Let $(M, J)$ be a Fano projective manifold. Can $(M, -J)$ be general type?


For complex curves and surfaces Kodaira dimension is diffeomorphism invariant so this cannot happen.



Fano manifold becoming general type upon conjugation
No. $(M,J)$ and $(M,-J)$ have conjugate pluri-canonical rings, hence have same Kodaira dimension.


*Proof.* Take a section $\mu$ of $K\_{(M, J)}^{\otimes n}$, then $\bar \mu$ is a holomorphic section of $K\_{(M,-J)}^{\otimes n}$. And vice versa.


