# There are many varieties with ample canonical bundle

Let $X$ be a smooth projective connected complex algebraic variety with ample canonical bundle. Let $h$ be the hilbert polynomial of the canonical bundle.

Why is the moduli stack of canonically polarized varieties with hilbert polynomial $h$ of dimension $>0$? (I know that it could have a zero-dimensional connected component, think of a fake projective plane.)

In other words, I'd like to know why the existence of $X$ forces the existence of many varieties that are "like" X.

• What makes you believe that there is an irreducible component of your moduli space of positive dimension? If you are asking whether or not that is true, I suggest editing your question a bit. – Jason Starr May 17 '13 at 12:27
• An irreducible compact quotient $X$ of a polydisc has ample canonical bundle but if its dimension is greater than one then $H^1(X,T_X)=0$, so that $X$ is rigid! See [Y. Matsushima and G. Shimura, "On the cohomology groups attached to certain vector valued differential forms on the product of the upper half planes". Ann. of Math. (2) 78 1963 417–449]. – diverietti May 17 '13 at 13:16
• Any canonically polarised surface of general type with the same Hilbert polynomial as a fake projective plane is a fake projective plane (and these are all rigid). – ulrich May 17 '13 at 14:21

If $X$ is a surface, then the Hilbert polynomial of the canonical divisor determines $d:=1-\chi(T_X)$ which is a lower bound for the dimension of any component of the moduli at the point $[X]$. Hence if $d>0$ it is true that the existence of $X$ forces the existence of other varieties like it.