Pure Hodge structure on the \(L_2\)-cohomology of varieties with isolated singularities (Q2710294)

Let \(V\) be a complex projective variety, and suppose its smooth part \(M\) is given the Kähler metric inherited from any imbedding in projective space. Then the forms on \(M\), square-integrable with respect to this metric, are independent of the choice of imbedding, and the cohomology \(H^*_2(V)\) of the resulting complex is called the \(L_2\)-cohomology of \(V\).NEWLINENEWLINENEWLINE\textit{J. Cheeger}, \textit{M. Goresky} and \textit{R. MacPherson} [in: Semin. Differential Geometry, Ann. Math. Stud. 102, 303-340 (1982; Zbl 0503.14008)] have conjectured that this cohomology is naturally isomorphic to \(IH^*(V)\), the intersection cohomology of \(V\) with middle perversity and that the classical properties (``Kähler package'') of ordinary cohomology which hold when \(V\) is smooth (pure Hoddge structure, Lefschetz decomposition) hold in general for the spaces \(H^*_2(V)\).NEWLINENEWLINENEWLINE\textit{T. Ohsawa} [Math. Z. 206, No. 2, 219-224 (1991; Zbl 0728.14022)] has proved the isomorphism \(H^*_2(V)\cong IH^*(V)\) when \(V\) has isolated singularities, and in this paper the authors exhibit the rest of the Kähler package in this case as well. One surprising subtlety is that ``boundary conditions'' on exterior derivatives, while not relevant to the definition of \(H^*_2(V)\), are needed to correctly specify the Hodge \((p,q)\)-components \(H^{p,q}_2(V)\), which are therefore eomplex algebraic invariants of \(V\). The arguments develop a series of delicate estimates beginning from a variation on a computation of \textit{H. Donnelly} and \textit{C. Fefferman} [Ann. Math., II. Ser. 118, 593-618 (1983; Zbl 0532.58027)].