Compact case (since you mention $\mathbf C^2/\mathbf Z^4$):

• For $M$ to be Kähler its first Betti number must be even, and conversely every compact complex surface with even $b_1(M)$ number is Kähler (Kodaira's conjecture, proved by Siu, Lamari, Buchdahl).

• Lamari and Buchdahl also describe "how many" Kähler metrics then exist, i.e. the so-called "Kähler cone" of classes in $H^{1,1}_{\mathbf R}(M)$ which can be represented by positive closed $(1,1)$-forms.

Compact case (since you mention $\mathbf C^2/\mathbf Z^4$):

• For $M$ to be Kähler its first Betti number must be even, and conversely every compact complex surface with even $b_1(M)$ is Kähler (Kodaira's conjecture, proved by Siu, Lamari, Buchdahl).

• Lamari and Buchdahl also describe "how many" Kähler metrics then exist, i.e. the so-called "Kähler cone" of classes in $H^{1,1}_{\mathbf R}(M)$ which can be represented by positive closed $(1,1)$-forms.