Let be a singular Hermitian complex space of pure dimension . We use a resolution of singularities to give a smooth representation of the --cohomology of -forms on . The central tool is an -resolution for the Grauert–Riemenschneider canonical sheaf . As an application, we obtain a Grauert–Riemenschneider-type vanishing theorem for forms with values in almost positive line bundles. If is a Gorenstein space with canonical singularities, then we also get an -representation of the flabby cohomology of the structure sheaf . To understand also the --cohomology of -forms on , we introduce a new kind of canonical sheaf, namely, the canonical sheaf of square-integrable holomorphic -forms with some (Dirichlet) boundary condition at the singular set of . If has only isolated singularities, then we use an -resolution for that sheaf and a resolution of singularities to give a smooth representation of the --cohomology of -forms.
"-theory for the -operator on compact complex spaces." Duke Math. J. 163 (15) 2887 - 2934, 1 December 2014. https://doi.org/10.1215/0012794-2838545