If an algebraic variety $X$ is proper, then the (naive) hodge numbers $h^{p,q}:= dim\, H^p(X, \Omega^q)$ are finite.



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E.g., finiteness of $h^{0,0}$ tells you that at least the variety is not (affine and not proper).


I am happy to assume X is smooth.



Does finiteness of hodge numbers imply properness?
Let $L$ be a degree $0$, nontorsion line bundle on an elliptic curve $E$, viewed as an affine scheme $X$ over $E$ with $\pi: X \to E$ the structure morphism. I claim all the Hodge numbers of $X$ are finite.


We have a canonical isomorphism $$\omega\_X = \pi^\* (\omega\_E \otimes L^{-1})$$ and short exact sequence $$0 \to \pi^\* \omega\_E \to \Omega^1\_X \to \pi^\* L^{-1}\to 0$$ that, together with the non-canonical isomorphism $\omega\_E=\mathcal O\_E$, reduce us to showing $H^p(X, \mathcal O\_X)$ and $H^p(X, \pi^\* L^{-1})$ are finite.


Because $\pi$ is affine, we have $$H^p(X, \mathcal O\_X) = H^p (E, \pi\_\* \mathcal O\_X)= H^p (E, \sum\_{n=0}^\infty L^{-\otimes n}) = \sum\_{n=0}^{\infty} H^p(E, L^{- \otimes n}) = H^p(E,\mathcal O\_X)$$ because as $L$ is nontorsion of degree $0$, all its powers have vanishing cohomology in every degree. Similarly $$H^p(X, \pi^\* L^{-1}) = H^p (E, \pi\_\* \pi^\* L^{-1}) = H^p (E, \sum\_{n=0}^{\infty} L^{-\otimes (n+1)}) = \sum\_{n=0}^\infty H^p(E, L^{- \otimes (n+1)}) = 0$$ by the same logic.




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In general, let $X$ be a variety with compactification $\overline{X}$ whose boundary is a simple normal crossings union of smooth divisors $D\_1,\dots D\_k$. If I calculated correctly, a sufficient condition for finiteness of Hodge numbers is that finitely many of the cohomology groups indexed by numbers $p,q$ and tuples $m\_1,\dots,m\_k$


$$H^p \left( \bigcap\_{ i | m\_i>0} D\_i, \Omega^q\_X \otimes N\_1^{\otimes m\_1} \otimes N\_2^{\otimes m\_2} \otimes \dots \otimes N\_k^{\otimes m\_k}\right)$$


are nonvanishing.


These should be the cohomology groups of the associated graded components of the pole order filtration of $\Omega^q\_X$, viewed as a quasicoherent sheaf on $\overline{X}$.


This condition explains my example when applied to the obvious compactification as a $\mathbb P^1$ bundle. However, it is not necessary, as can be seen by simply choosing another compactification.


This vanishing might be an interesting condition to study. There are some explicit constructions generalizing my example (for $A$ an abelian variety and $X$ any variety, $L\_1$ a very general degree $0$ line bundle on $A$ and $L\_2$ any line bundle on $X$, a smooth divisor $X \times A$ with normal bundle $L\_1 \otimes L\_2$ does the job.)


The only necessary conditions I found were a bunch of vanishing characteristic class integrals (I think the integral of any nonconstant monomial in $c\_1(D\_1),\dots,c\_1(D\_k)$ times the Chern character of $\Omega^q$ times the Todd class must vanish) which don't really say much about the structure of the $D\_i$.


