Log–canonical forms and log canonical singularities

Abstract For a normal subvarietyVof ℂnwith a good ℂ*–action we give a simple characterization for when it has only log canonical, log terminal or rational singularities. Moreover we are able to give formulas for the plurigenera of isolated singular points of such varieties and of the logarithmic Kodaira dimension ofV\{0}. For this purpose we introduce sheaves ofm–canonical andL2,m–canonical forms on normal complex spaces. For the case of affine varieties with good C*–action we give an explicit formula for these sheaves in terms of the grading of the dualizing sheaf and its tensor powers.