Hodge结构在研究复射影代数流形中具有非常核心的地位。诸多上同调消灭定理（如Kodaira-Nakano消灭定理、Kawamata-Viehweg消灭定理等）均来自于Hodge理论。如今奇异代数簇的研究越来越受到关注。尤其关于相交复形下推出的分解定理在吴宝珠（Ngô Bảo Châu）对基本引理的证明（2010年菲尔兹奖工作），张伟、恽之玮在L函数的Taylor展开工作（2018年数学新视野奖工作）中都起到了核心作用。然而人们对相交上同调的Hodge结构的知识还十分有限。本项目旨在利用L2上同调方法研究奇异代数簇上相交上同调Hodge结构，并进一步思考更大框架Hodge module下的L2上同调表示问题。

Hodge Structure is crucial in the study of complex projective smooth varieties. Various cohomological vanishing theorems (Kodaira-Nakano vanishing, Kawamata-Viehweg vanishing, etc) follow from the theory of Hodge structure. Nowadays singular varieties attract more and more attentions. In particular, the decomposition theorem of the the pushforward of the intersection complex plays crucial role in Ngo's proof of the fundamental lemma (2010 Fields medal) and in the work of the Taylor expansion of the L-functions by W. Zhang and Y. Z. Wei (2018 New Horizons In Mathematics Prize). However, the knowledge of the Hodge structure on the intersection cohomology is limited. In this project we aim to study this Hodge structure by using L2 cohomologies. Moreover we shall study the L2 representation of the Hodge structures of Hodge modules.