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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.

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2 Answers 2

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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).

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Ziegler's answer and yours complements each other in the following way. If our compact complex manifold admits any Kahler metric ever, then on can find all the metrics by first fixing a Kahler class in the Kahler cone and then changing the metric in the same class by choosing a smooth function. –  Asghar Ghorbanpour Aug 14 at 12:41

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.

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