Von Neumann dimension, Hodge index theorem and geometric applications

Abstract

This note contains a reformulation of the Hodge index theorem within the framework of Atiyah's L 2index theory. More precisely, given a compact Kähler manifold (M, h) of even complex dimension 2m, we prove thatwhere σ(M ) is the signature of M and h p,q (2),Γ (M ) are the L 2 -Hodge numbers of M with respect to a Galois covering having Γ as group of Deck transformations. Likewise we also prove an L 2 -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 L 2 -Hodge numbers.