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\(L^ 2\)-cohomology of normal algebraic surfaces. I. (English) Zbl 0627.14016
The authors show that the \(L^ 2\)-cohomology of the non-singular part of a complex projective surface with the induced Fubini metric is isomorphic to the middle perversity intersection homology. This was conjectured by J. Cheeger, M. Goresky and R. MacPherson [Semin. differ. Geom., Ann. Math. Stud. 102, 303-340 (1982; Zbl 0503.14008)], and is proved using a modification of arguments of J. Cheeger [in Geometry of the Laplace operator, Honolulu/Hawaii, Proc. Symp. Pure Math. 36, 91-146 (1980; Zbl 0461.58002)].
Reviewer: P.E.Newstead

MSC:
14F30 \(p\)-adic cohomology, crystalline cohomology
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
55N35 Other homology theories in algebraic topology
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References:
[1] Cheeger, J.: On the Hodge theory of Riemannian pseudo-manifolds. Proc. Symp. Pure Math. AMS36, 91-146 (1980) · Zbl 0461.58002
[2] Cheeger, J., Goresky, M., MacPherson, R.:L 2-cohomology and intersection homology for singular algebraic varieties. Proc. of Year in Differential Geometry I.A.S.Yau (ed.). Ann. Math. Stud.102, 303-340 (1982) · Zbl 0503.14008
[3] Goresky, M., MacPherson, R.: Intersection homology theory. Topology19, 135-162 (1980) Intersection homology II. Invent. Math.72, 77-130 (1983) · Zbl 0448.55004
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