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

Xuanming Ye, Kang Zuo

Number 35
Authors Xuanming Ye
Kang Zuo
Year 2011

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)).

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