One the one hand, any topological surface has a complex structure, so for any natural number $g$, there exists a complex curve with genus $g$. On the other hand, we have a Chow's theorem, which says that any complex analytic variety is algebraic. This means that there exists a non-singular algebraic curve with genus $g$. But algebraic curves have a well-defined degree $d$ which connect to the notion of genus using the formula $g=(d-1)(d-2)/2$. This suggests that $g$ cannot be any natural number.
Obviously this argument has a flaw, but where?