Since is given by equations, we can interpret it as a map of complex manifolds.

It is especially popular in the context of complex manifolds.

It is a complex manifold of complex dimension 3g 3.

Often the space is a complex manifold and the group is a discrete group.

It follows from this definition that an almost complex manifold is even dimensional.

Let X be a complex algebraic manifold and a good compactification.

See complex differential form and almost complex manifold for applications of these ideas.

Note however that it is very hard to decide whether a complex manifold is kähler or not.

Similarly, the quaternionic analogs of these are also complex manifolds.

A complex manifold admitting a positive line bundle is kähler.