局部单值拟投射流形基本群约化表示的$L^2$与交上同调
$L^2$ and intersection cohomologies for the reductive representation of the fundamental groups of quasiprojective manifolds with unipotent local monodromy
摘要：
设$X$是一个射影流形，$D$是$X$的一个正规交叉因子。根据Jost-Zuo定理，如果我们有一个局部单值的基本群$π_{1}(X^{*})$的约化表示$ρ$，其中$X^*=X-D$，那么在与$X^*$上的局部系统$\mathbb V$相关联的平坦丛$\mathcal V$上存在一个驯服的多调和度量$h$.因此，我们得到一个调和丛$(E, θ, h)$，其中$θ$是Higgs场，即满足$θ^2=0$的$End(E)\otimesΩ^{1,0}_{X^*}$的全纯截面.本文研究了$X^*$上的调和丛$(E,θ,h)$.我们将证明交上同调$IH^{k}(X; \mathbb V)$同构于$L^{2}$ -上同调$H^{k}(X, (\mathcal A_{(2)}^{\cdot}(X,\mathcal V), \mathbb D))$ .
英文摘要：
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 $ρ$ of the fundamental group $π_{1}(X^{*})$ 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 $ρ$ over $X^*$. Therefore, we get a harmonic bundle $(E, θ, h)$, where $θ$ is the Higgs field, i.e. a holomorphic section of $End(E)\otimesΩ^{1,0}_{X^*}$ satisfying $θ^2=0$. In this paper, we study the harmonic bundle $(E,θ,h)$ 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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