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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)].
 14F30 $$p$$-adic cohomology, crystalline cohomology 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 55N35 Other homology theories in algebraic topology