I am sorry if this is obvious, but I needed clarify few things about complex torus.

In two dimensions the group of large diffeomorphisms on complex torus are defined by the modular group, whose elements are represented by the elements of \(SL(2, \mathbb{Z})\). I understand that given two complex tori defined by the complex vectors:\((\omega_1, \, \omega_2 )\) and \((\omega'_1,\, \omega'_2)\),we can infer that they are equivalent if the vectors can be obtained from one other via modular transformation.

1. Suppose we have a non-contractable loop on the torus with the winding numbers \(m\) and \(n\). What exactly happens to them under a modular transformation? Do they change the same way as \((\omega_1, \, \omega_2 )\)?

2. One of the results of KAM theorem is that quasiperiodic trajectories on the real plane share the same topology with a torus. Can we make a similar statement if the trajectories live in the complex plane i.e do complex quasiperiodic orbits have the topology of a complex torus?