In the Wikipedia of Arithmetic genus, for a complex projective manifold of complex dimension 1, the arithmetic genus satisfies $\chi=1-g$.
I also see the relation $\chi=2-2g$ for a connected, orientable surface (Genus(mathematics)-Wikipedia).
Are the Euler characteristics the same and the arithmetic genus is different from the genus? Or the Euler characteristics are different, and there is only one genus?
Thanks in advance.