NOVELTY - Manufacture of positive electrode active material involves mixing nickel-cobalt-manganese composite hydroxide particles (I) and lithium compound, such that ratio of the number of lithium atoms to the sum total of the number of atoms of the metallic element is 0.95-1.20, obtaining lithium mixture, increasing the temperature of the obtained lithium mixture to 30-800 degrees C at a rate of 2-10 degrees C/minute in oxidative atmosphere, increasing the temperature to 800-1000 degrees C for 1-5 hours and baking the resultant lithium mixture for 3-8 hours.    USE - Manufacture of positive electrode active material used for nonaqueous electrolyte secondary battery (all claimed). Uses include but are not limited to mobile telephone, notebook computer, electric vehicle and hybrid vehicle.    ADVANTAGE - The method provides positive electrode active material having excellent stability and handleability. The nonaqueous electrolyte secondary battery containing the positive electrode active material has excellent output characteristics even under cryogenic environment of 0 degrees C-(-30) degrees C.    DETAILED DESCRIPTION - Manufacture of positive electrode active material involves mixing nickel-cobalt-manganese composite hydroxide particles of formula: NixCoyMnzMt(OH)2+a (I) (where a is 0-0.5, x+y+z+t is 1, x is 0.3-0.7, y and z are 0.1-0.4, t is 0.0003-0.05 and M is magnesium, calcium, aluminum, titanium, vanadium, chromium, zirconium, niobium, molybdenum, hafnium, tantalum or tungsten) and lithium compound, such that ratio of the number of lithium atoms to the sum total of number of atoms of the metallic element is 0.95-1.20, obtaining lithium mixture, increasing the temperature of the obtained lithium mixture to 30-800 degrees C at a rate of 2-10 degrees C/minute in oxidative atmosphere, increasing the temperature to 800-1000 degrees C for 1-5 hours and baking the resultant lithium mixture for 3-8 hours. INDEPENDENT CLAIMS are included for the following:    (1) positive electrode active material, which contains hexagonal lithium-nickel-cobalt-manganese complex oxide particles of formula: Li1+sNixCoyMnzMtO2, where s is (-0.05)-0.20; and    (2) nonaqueous electrolyte secondary battery.