The present invention discloses a CMP abrasive comprising cerium oxide particles, a dispersant, an organic polymer having an atom or a structure capable of forming a hydrogen bond with a hydroxyl group present on a surface of a film to be polished and water, a method for polishing a substrate comprising polishing a film to be polished by moving a substrate on which the film to be polished is formed and a polishing platen while pressing the substrate against the polishing platen and a polishing cloth and supplying the CMP abrasive between the film to be polished and the polishing cloth, a method for manufacturing a semiconductor device comprising the steps of the above-mentioned polishing method, and an additive for a CMP abrasive comprising an organic polymer having an atom or a structure capable of forming a hydrogen bond with a hydroxyl group present on a surface of a film to be polished, and water.