Pure Hodge structure on the L₂-cohomology of varieties with isolated singularities

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Let V be a complex projective variety with isolated singularities. Let the smooth part be given the metric induced by a projective imbedding. Then we develop the L₂ harmonic theory and construct a pure Hodge structure on the L₂-cohomology of V. If the dimension of V is two, we put a cohomological Hodge structure on its complex of sheaves of L₂ forms.