# Why is the Hodge-Deligne polynomial a polynomial?

Let $X$ be a compact complex manifold. Its Hodge-Deligne polynomial is then defined to be $\sum_{p, q \geq 0} (-1)^{p+q} h^{p, q}(X)$ where $h^{p, q}(X):= \mbox{dim}_{\mathbb{C}}H^{p, q}(X)$.

The question now is: why is this a polynomial? More concretely I want to know the following:

• Why are the numbers $h^{p, q}(X)$ always finite, or equivalently the vector spaces $H^{p, q}(X)$ finite-dimensional.

• Why are the values $h^{p, q}(X)$ equal to $0$ if $p+q$ is large enough?

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