Chern/Hodge numbers of the conjugate complex manifold

Question

Let $(M, J)$ be a complex projective manifold. Can $(M, -J)$ have different Chern/Hodge numbers?

Answer 1

These are all the same.

As for Hodge numbers, you can choose a Kahler metric $g$ on $(M,J)$, and it will also be Kahler for $(M,-J)$. Now we know that $h^{p,q}$ is the dimension of the space of harmonic $(p,q)$-forms. A harmonic $(p,q)$ form for $(M,g,J)$ gives you a harmonic $(q,p)$ form on $(M,g,-J)$. And since $h^{p,q}=h^{q,p}$ for a Kahler manifold, we are done.

Concerning Chern numbers, note that the tangent bundles $(M,J)$ and $(M,-J)$ are dual. For dual bundles $V$ and $V^*$ we know that $c_k(V)=(-1)^kc_k(V^*)$. So a product of Chern classes on $V$ and $V^*$ is the same if it is in degree divisible by $4$ and it differs in sign if it is in degree $4k+2$. It remains to note that orientations the of $(M,J)$ and $(M,-J)$ coincide when the complex dimension of $M$ is even, and differ when it is odd.