NOVELTY - Manufacture of positive electrode active material involves mixing compound (A) and compound (B), baking at 450-700 degrees C in oxygen-containing atmosphere, further baking the resultant baked product in oxygen-containing atmosphere and obtaining complex oxide (I). The compound (A) is carbonic acid compound containing nickel, cobalt and/or manganese having diameter (D90) of less than 40 mu m and volume-based diameter (D50) of 3-15 mu m, measured by laser diffraction scattering-type particle size distribution analyzer. The compound (B) is lithium carbonate.    USE - Manufacture of positive electrode active material used for positive electrode for lithium-ion secondary battery (all claimed) for mobile telephone, notebook computer and vehicle.    ADVANTAGE - The lithium-ion secondary battery provided with the positive electrode obtained using the positive electrode active material has excellent initial stage charging and discharging efficiency and initial stage discharge capacity.    DETAILED DESCRIPTION - Manufacture of positive electrode active material involves mixing compound (A) and compound (B), baking the resultant mixture at 450-700 degrees C in oxygen-containing atmosphere, further baking the resultant baked product at 750-1000 degrees C in oxygen-containing atmosphere and obtaining complex oxide (I). The compound (A) is carbonic acid compound containing nickel, cobalt and/or manganese having diameter (D90) of less than 40 mu m and volume-based diameter (D50) of 3-15 mu m, measured by laser diffraction scattering-type particle size distribution analyzer by making water into a dispersion medium. The compound (B) is lithium carbonate having diameter (D90) of less than 40 mu m and volume-based diameter (D50) of 3-15 mu m, measured by laser diffraction-scattering type particle size distribution analyzer by making 2-propanol into dispersion medium. The molar ratio of lithium is more than 1.2 times with respect to the total molar content of nickel, cobalt and manganese, nickel and cobalt and/or manganese and lithium in the complex oxide (I). INDEPENDENT CLAIMS are included for the following:    (1) positive electrode, which comprises positive electrode active material and binder; and    (2) lithium-ion secondary battery, which comprises positive electrode, negative electrode and nonaqueous electrolyte.