A CMP abrasive comprising a cerium oxide slurry containing cerium oxide particles, a dispersant and water, and a liquid additive containing a dispersant and water; and a liquid additive for the CMP abrasive. A method for polishing a substrate which comprises holding a substrate having, formed thereon, a film to be polished against a polishing pad of a polishing platen, followed by pressing, and moving the substrate and the polishing platen while supplying the above CMP abrasive in between the film to be polished and the polishing pad to thereby polish the film to be polished. The CMP abrasive and the method for polishing can be used for polishing a surface to be polished such as a silicone oxide film or a silicon nitride film without contaminating the surface to be polished with an alkali metal such as sodium ions and with no flaws, and the CMP abrasive is excellent in storage stability.