MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.