Inequivalent holomorphic atlases

Question

What is an example of a differential manifold which possesses two holomorphic atlases which are equivalent as differential atlases but inequivalent as holomorphic atlases? Moreover, for this example, are these nice properties distinguishing each atlas?

Answer 1

In dimensions smaller than 4, there is only one differential structure for each topological manifold. That was proved by Tibor Radó for dimension $1$ and $2$, and by Edwin E. Moise in dimension $3$.

Now take $T_g$, the connected sum of $g$ tori: this is a smooth topological surface of genus $g$, hence there is a unique smooth structure on it. By contrast, if $g \ge 2$ then the complex structures on $T_g$ (up to the action of homeomorphisms that are isotopic to the identity) form a space of real dimension $6g-6$, see Teichmuller theory.