\(L_ 2\)-cohomology and intersection homology of certain algebraic varieties with isolated singularities (Q1087601)

Let V be an algebraic surface with isolated singularities, let \(\tilde V\) be a resolution such that the exceptional divisor E has normal crossings. The author proves the existence of a complete Kähler metric on \(\tilde V\setminus E\) (hence on \(V\setminus \sin gularities)\), whose local models near a smooth point and a crossing point of E are the quotient of a 2- ball and a Hilbert modular surface respectively. The actual constructions of the metric is rather complicated and so is the calculation of the corresponding \(L_ 2\)-cohomology of \(U\setminus E\) for a neighbourhood U of E in \(\tilde V\) using a spectral sequence argument. Finally the author combines the result of these calculations with the sheaf theoretic characterization of intersection homology in order to prove: The \(L_ 2\)-cohomology of the complete Kähler metric on \(V\setminus \sin gularities\) is isomorphic to the (middle) intersection homology of V in the complementary dimensions. In a note added in proof the author claims that the results of this paper hold for all varieties (any dimension) with isolated singularities. Similar results have been obtained by \textit{T. Ohsawa} [''Hodge spectral sequence on compact Kähler spaces'', Publ. Res. Inst. Math. Sci. 23, 265-274 (1987)].