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\(L^ 2\) and intersection cohomologies for a polarizable variation of Hodge structure. (English) Zbl 0611.14006

The authors consider a polarized variation of Hodge structure (\({\mathcal V},{\mathcal V}_{{\mathbb{Z}}},{\mathcal S},{\mathcal F})\) of weight k over a complex manifold \(X\subset \bar X\), where \({\mathcal V}\) is a locally constant sheaf of finite-dimensional complex vector spaces, \({\mathcal V}_{{\mathbb{Z}}}\subset {\mathcal V}\) is a sheaf of lattices, F is a decreasing filtration of \({\mathcal O}_ X\otimes_{{\mathbb{C}}}{\mathcal V}\) by locally free \({\mathcal O}_ X\)- modules \({\mathcal F}^ p\), and \(\bar X\) is a compact Kähler manifold containing X as a Zariski open subset such that \(\bar X\setminus X\) is a divisor with normal crossings. In the case when \(X=\bar X\) is compact, Deligne has constructed canonical Hodge structures of weight \(p+k\) on the cohomology groups \(H^ p(X,{\mathcal V})\) [cf. S. Zucker, Ann. Math., II. Ser. 109, 415-476 (1979; Zbl 0446.14002)]. The authors show that in general the \(L^ 2\)-cohomology groups \(H^*_{(2)}(\bar X,V)\) carrying canonical Hodge structures coincide with the intersection cohomologies \(IH^*(\bar X,{\mathcal V})\). Thus the intersection cohomology groups \(IH^ p(\bar X,{\mathcal V})\) carry canonical Hodge structures of weight \(p+k\). This proves a conjecture of Deligne [another proof was announced by M. Kashiwara and T. Kawai, Proc. Japan Acad., Ser. A 61, 164-167 (1985; Zbl 0576.14010)].
Reviewer: F.L.Zak

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14F99 (Co)homology theory in algebraic geometry
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References:

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