Von Neumann dimension, Hodge index theorem and geometric applications

This note contains a reformulation of the Hodge index theorem within the framework of Atiyah's $L^2$-index theory. More precisely, given a compact K\"ahler manifold $(M,h)$ of even complex dimension $2m$, we prove that $$\sigma(M)=\sum_{p,q=0}^{2m}(-1)^ph_{(2),\Gamma}^{p,q}(M)$$ where $\sigma(M)$ is the signature of $M$ and $h_{(2),\Gamma}^{p,q}(M)$ are the $L^2$-Hodge numbers of $M$ with respect to a Galois covering having $\Gamma$ as group of Deck transformations. Likewise we also prove an $L^2$-version of the Fr\"olicher index theorem. Afterwards we give some applications of these two theorems and finally we conclude this paper by collecting other properties of the $L^2$-Hodge numbers.