NOVELTY - Complex oxide contains specific composition containing lithium, nickel, manganese and cobalt. The primary average particle diameter of the complex oxide is 3-20 microns.    USE - For structural layer of anode material of lithium secondary battery (all claimed) used as power supply for notebook personal computer, pen input personal computer, mobile personal computer, electronic book player, mobile telephone, portable facsimile, portable copier, portable printer, headphone stereo, video movie, liquid crystal television, handy cleaner, transceiver, electronic notebook, calculator, memory card, portable tape recorder and radio, and as backup power supply for motor, luminaire, toy, game machine device, clock, electronic flash, camera, electric bicycle and electric motor scooter.    ADVANTAGE - The lithium-nickel-manganese-cobalt complex oxide has high bulk density and provides anode capable of improving cycle characteristics of lithium secondary battery.    DETAILED DESCRIPTION - The lithium-nickel-manganese-cobalt complex oxide contains composition of formula: Li1+xNi1-y-z-pMnyCozMpO2, where x and p are 0-0.2, y and z are 0.1-0.5 such that y+z+p is 0.2-0.8 and M is aluminum, iron, titanium, magnesium, chromium, gallium, copper, zinc or niobium. The primary average particle diameter of the complex oxide is 3-20 microns.    INDEPENDENT CLAIMS are included for:    (a) manufacture of the lithium-nickel-manganese-cobalt complex oxide, which involves baking a precursor mixture containing lithium compound, nickel compound, manganese compound, cobalt compound and compound of aluminum, iron, titanium, magnesium, chromium, gallium, copper, zinc or niobium in non-oxidizing gas atmosphere at 700degreesC or more, and baking obtained mixture containing Li6Co1-aQaO4 phase (where Q is nickel and/or manganese, a is 0-0.8) in oxygen-containing gas atmosphere;    (b) anode, which has anode active material layer containing the lithium-nickel-manganese-cobalt complex oxide and binder on a collector; and    (c) lithium secondary battery, which has the anode, cathode capable of occluding and discharging lithium and non-aqueous electrolyte containing lithium salt    .