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

TL;DR

This paper investigates the relationship between intersection cohomology and $L^2$-cohomology for harmonic bundles arising from reductive representations with unipotent monodromy on quasiprojective manifolds.

Contribution

It proves an isomorphism between intersection cohomology and $L^2$-cohomology for such harmonic bundles, extending previous understanding of their cohomological properties.

Findings

Intersection cohomology $IH^{k}(X; ext{V})$ is isomorphic to $L^{2}$-cohomology $H^{k}(X, ( ext{A}_{(2)}^{ullet}(X, ext{V}), ext{D}))$.

Harmonic bundles with unipotent monodromy admit a tame pluriharmonic metric, enabling this cohomological comparison.

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 $H^{k}(X, (\mathcal A_{(2)}^{\cdot}(X,\mathcal V), \mathbb D))$.