Some mixed Hodge structures on $$l^{2}$$ -cohomology groups of coverings of Kähler manifolds

Abstract

We give methods to compute l 2 −cohomology groups of a covering manifold obtained by removing the pullback of a (normal crossing) divisor to a covering of a compact Kähler manifold.We prove that in suitable quotient categories, these groups admit natural mixed Hodge structure whose graded pieces are given by the expected Gysin maps.

Application of Cheeger-Gromov theory to the $l^2$-cohomology of harmonic Higgs bundles over covering of finite volume complete manifolds

“…1) The more traditional proof using first a ∂−primitive, then a ∂ * −primitive was given in [30,29]. However, it requires uniform Sobolev spaces to justify integrations by parts.…”

“…Definition 7.2.5. (see [12,30,35]) i) Assume that (E, h, ∇) = p * (E ′ , h ′ , ∇ ′ ) is the pullback of a bundle with a flat connection on Ȳ . Let E ′ → Ȳ be the local system it defines, and let p * (2) (E ′ ) → Ȳ be its l 2 −direct image sheaf.…”

“…On a compact Kähler manifold, the ∂∂−lemma is fundamental for the development of Hodge theory ( [79,27]), and Mixed Hodge theory of Deligne ([25, 26]). Applications in the context of l 2 −Hodge theory were given in [30,29].…”

A Factorization Theorem for Curves with Vanishing Self-Intersection

“…The fact that M(Γ) admits a finite trace implies that it is enough to be injective or with dense range. We will use the Von Neumann algebra M(Γ) through the following lemma ( [11] cor.3.4.6): Proof. From 3.2.1, we may assume C connected and effective.…”

“…, which is a non vanishing current (for λ(f ) is injective). The degeneracy of U(Γ)−spectral sequence implies that the (1, 0)−part of this class is represented by a square integrable logarithmic one form L. From [11], [10], we know that L is closed.…”