# 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?)

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By Kodaira-Spencer theory, you can compute the space of first-order deformations of the complex structure as a cohomology group. For a fixed choice of complex structure, this space is (more or less) the tangent space of the moduli space of complex structures at the point that describes the chosen complex. structure. –  Scott Carnahan Aug 1 '11 at 15:06
Any references about this? And I still have no idea how to put all the complex structure together.(Its topology? Manifold structure?) –  Honglu Aug 2 '11 at 3:55
I do not know the answer, but i can add that in general such a moduli space need not be a manifold. In the case of a torus (which is the edge of my knowledge) it is an "orbifold", which is roughly supposed to be a manifold with corners. To see a picture, see en.wikipedia.org/wiki/…, but glueing around the edges in some way is required.. –  Joachim Sep 23 '12 at 17:21
the space of all complex structure on a compact complex manifold is an infinite dimensional complex manifold, with the $L^2$ norm defined by using the Beltrami differenctial, it can be shown that this infinite dimensional space is actually kahler, but i can not see how it could be finite dimensional. –  Vicky Cheung Sep 24 '13 at 2:06