Pure Hodge structure of the harmonic \(L^ 2\)-forms on singular algebraic surfaces (Q922596)

Let X be a complex algebraic surface imbedded in \(P^ N({\mathbb{C}})\) and denote by S the set of singular points of X. The author proves that the harmonic \(L^ 2\)-spaces on X-S have pure Hodge and hard Lefschetz structures, provided S consists of isolated points. The Kähler metric used to define the harmonic \(L^ 2\)-spaces is the one induced from the Fubini-Study metric on \(P^ N({\mathbb{C}})\). The pure Hodge structure is obtained for \(i=p+q\neq 2\) and is conjectured for \(i=p+q=2\).