Why the moduli space of complex structure in a compact complex manifold is of finite dimension I admit the statement in the title might be much too unclear. I just heard from my teacher that we can form a finite dimension moduli space of all the complex structure in a compact complex manifold. He said it is a standart fact in complex geometry but I failed to find out both the correct statement and the proof. So I'm asking for both a theorem you think is closest to this kind of meaning and a reference of its proof. Thank everyone in advance. (Should I add some more tags for such kind of vague question?)
 A: I just asked this question to Siu at Harvard. He responded immediately that Brieskorn gave an infinite diml example in math annalen 1965.
Deformation theory implies the space is locally finite dimensional, but the local dimension goes to infinity in Brieskorn's example.Dennis Sullivan
Here is Yum Tong Siu's precise response:
If a complex structure is required for the space of complex structures, one can only use Kuranishi's semi-universal moduli space.  For such a setup, there is the following situation.
One starts out with a compact complex algebraic manifold X_1 and forms its semi-universal moduli space M_1 whose dimension at the point corresponding to X_1 is d_1.
Along M_1 the manifold X_1 is deformed to X_2 whose semi-universal moduli space M_2 has dimension d_2 at the point corresponding to X_2.
Then one continues the process and gets d_{j+1}>d_j to end up with a topologically connected component whose analytic branches have dimension going to infinity.
A concrete example for this situation is Briskorn's generalization of Hirzebruch surfaces in Math. Annalen 157 (1965), 343-357.
The relevant statements in Brieskorn's 1965 paper are:
Satz 2.6 on p.349
Satz 6.1 i) on p.354
Satz 6.2 on p.355.
Yum Tang Siu.
