$L^2$ and intersection cohomologies for the reductive representation of the fundamental groups of quasiprojective manifolds with unipotent local monodromy

Abstract

Let $X$ be a projective manifold, and $D$ be a normal crossing divisor of $X$. By Jost-Zuo's theorem that if we have a reductive representation $\rho$ of the fundamental group ${\pi }_1(X^{\ast })$ with unipotent local monodromy, where $X^{\ast }=X-D$, then there exists a tame pluriharmonic metric $h$ on the flat bundle $\mathcal{V}$ associated to the local system $\mathbb{V}$ obtain from $\rho$ over $X^{\ast }$. Therefore, we get a harmonic bundle $(E,\theta ,h)$, where $\theta$ is the Higgs field, i.e. a holomorphic section of $End(E)\otimes {\Omega }_{X^{\ast }}^{1,0}$ satisfying ${\theta }^2=0$. In this paper, we study the harmonic bundle $(E,\theta ,h)$ over $X^{\ast }$. We are going to prove that the intersection cohomology $I{H}^k(X;\mathbb{V})$ is isomorphic to the $L^2$-cohomology $H^k(X,({\mathcal{A}}_{(2)}^{\cdot }(X,\mathcal{V}),\mathbb{D}))$.

This paper has been withdrawn by the author due to a crucial sign error in lemma 3.7