Von Neumann dimension, Hodge index theorem and geometric applications

A reformulation of the Hodge index theorem within the framework of Atiyah’s L2-index theory is provided. More precisely, given a compact Kähler manifold (M, h) of even complex dimension 2m, we prove that σ(M)=∑p,q=02m(-1)ph(2),Γp,q(M)where σ(M) is the signature of M and h(2),Γp,q(M) are the L2-Hodge numbers of M with respect to a Galois covering having Γ as group of deck transformations. Likewise we also prove an L2-version of the Frölicher index theorem, see (3). Afterwards we give some applications of these two theorems and finally we conclude this paper by collecting other properties of the L2-Hodge numbers.