# Are there only finitely varieties of general type dominated by a given variety in the following sense

Let $X$ be a smooth projective complex algebraic variety. I believe the answer to the following question should be well-known. It is an instance of the Iitaka conjecture.

There are only finitely many isomorphism classes of smooth projective varieties of general type $Y$ such that there exists a surjective proper birational morphism from $X$ to $Y$.

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The answer is yes. You can look at this paper: Guerra, Pirola, On the finiteness theorem for rational maps on a variety of general type, Collect. Math. 60 (2009), no. 3, 261–276.

It contains an overview of the topic, and besides it gives some effective bounds for the number of such maps.

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