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

Research paper by **Xuanming Ye, Kang Zuo**

Indexed on: **11 Jul '12**Published on: **11 Jul '12**Published in: **Mathematics - Differential Geometry**

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### 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 $\pi_{1}(X^{*})$ with unipotent local monodromy, where
$X^*=X-D$, 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
$X^*$. Therefore, we get a harmonic bundle $(E, \theta, h)$, where $\theta$ is
the Higgs field, i.e. a holomorphic section of
$End(E)\otimes\Omega^{1,0}_{X^*}$ satisfying $\theta^2=0$.
In this paper, we study the harmonic bundle $(E,\theta,h)$ over $X^*$. We are
going to prove that the intersection cohomology $IH^{k}(X; \mathbb V)$ is
isomorphic to the $L^{2}$-cohomology $H^{k}(X, (\mathcal
A_{(2)}^{\cdot}(X,\mathcal V), \mathbb D))$.