Let $M$ be a complex manifold of complex dimension 2. What do we know about the set all Kähler metrics on $M$ in general and in the case of 4-torus $C^2/Z^4$? 


For the case of surfaces ($dim\_C=1$), any compatible metric is Kähler and by the uniformization theorem, the answer is that every two such metrics are conformally equivalent and the set all Kähler metrics is nonempty. 



All Kähler metrics on a complex manifold?
Let $M$ be a compact complex manifold. The set of Kahler metric on $M$ in a given Kahler class is parametrized by the set of positive volume forms with given integral, because (by Calabi-Yau theorem) any given positive volume form is a volume form of a Kahler metric in a given cohomology class, assuming their integrals agree. This is actually used when they put a structure of an infinite-dimensional symmetric space on the space of all Kahler metrics. See for example here: <http://www.emis.de/journals/NYJM/JDG/p/2000/56-2-1.pdf> (THE SPACE OF KAHLER METRICS, by XIUXIONG CHEN).


